Global Regularity for General Non-Linear Wave Equations I ...bdriver/DRIVER/Miscellaneous... ·...

Global Regularity for General Non-Linear WaveEquations I (6 + 1) and Higher Dimensions

Jacob Sterbenz

Department of Mathematics Princeton University Princeton

New Jersey USA

ABSTRACT

Following work of Tataru [Tataru D (1998) Local and global results for wavemaps I Comm Partial Differential Equations 23(9ndash10)1781ndash1793 Tataru D(1999) On the equation ampu frac14 jHuj2 in 5thorn 1 dimensions Math Res Lett 6

(5ndash6)469ndash485] we solve the division problem for wave equations with genericquadratic non-linearities in high dimensions Specifically we show that non-linearwave equations which can be written as systems involving equations of the form

ampf frac14 fHf andampf frac14 jHfj2 are well-posed with scattering in eth6thorn 1THORN and higherdimensions if the Cauchy data are small in the scale invariant lsquo1 Besov space _BBsc1This paper is the first in a series of works where we discuss the global regularity

properties of general non-linear wave equations for all dimensions 4 n

Key Words Semi-linear wave equation Strichartz estimates

1991 Mathematics Subject Classification 35L70

Correspondence Jacob Sterbenz Department of Mathematics Princeton UniversityPrinceton NJ 08544 USA E-mail sterbenzmathprincetonedu

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS

Vol 29 Nos 9 amp 10 pp 1505ndash1531 2004

1505

DOI 101081PDE-200037764 0360-5302 (Print) 1532-4133 (Online)

Copyright 2004 by Marcel Dekker Inc wwwdekkercom

4_LPDE29_09amp10_R3_102804

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1 INTRODUCTION

In this paper our aim is to give a more or less complete description of the globalregularity properties of generic homogeneous quadratic semi-linear wave equationson eth6thorn 1THORN and higher dimensional Minkowski space The equations we will considerare all of the form

ampf frac14 NethfDfTHORN eth1THORN

Here amp frac14 2t thorn Dx denotes the standard wave operator on Rnthorn1 and N is a

smooth function of f and its first partial derivatives which we denote by Df Forall of the nonlinearities we study here N will be assumed to be at least quadraticin nature that is

NethX Y THORN frac14 OethjethX Y THORNj2THORN ethX Y THORN 0

The homogeneity condition we require N to satisfy is that there exist a (vector) ssuch that

Nethlsf lsthorn1DfTHORN frac14 lsthorn2NethfDfTHORN eth2THORN

where we use multiindex notation for vector N The condition (2) implies thatsolutions to the system (1) are invariant (again solutions) if one performs the scaletransformations

fethTHORNff lsfethlTHORN eth3THORN

The general class of equations which falls under this description contains virtually allmassless non-linear field theories on Minkowski space including the Yang Millsequations (YM) the wavendashmaps equations (WM) and the MaxwellndashDirac equations(MD) We list the schematics for these systems respectively as

ampA frac14 ADAthorn A3 ethYMTHORNampf frac14 jDfj2 ethWMTHORNampu frac14 ADu ethMDTHORNampA frac14 jDuj2

The various values of s for these equations are (respectively) s frac14 1 s frac14 0 ands frac14 eth12 1THORN For a more complete introduction to these equations see for instancethe works Foschi and Klainerman (2000) and Bournaveas (1996) For the purposesof this paper will will only be concerned with the structure of these equations at thelevel of the generic schematics (YM)ndash(MD)

The central problem we will be concerned with is that of giving a precisedescription of the regularity assumptions needed in order to guarantee that the

1506 Sterbenz

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Cauchy problem for the system (1) is globally well posed with scattering (GWPS)That is given initial data

feth0THORN frac14 f tfeth0THORN frac14 g eth4THORN

we wish to describe how much smoothness and decay ethf gTHORN needs to possess in orderfor there to exist a unique global solution to the system (1) with this given initialdata We also wish to show that the solutions we construct depend continuouslyon the initial data and are asymptotic to solutions of the linear part of (1) We willdescribe shortly in what sense we will require these notions to hold

Our main motivation here is to be able to prove global well-posedness for non-linear wave equations of the form (1) in a context where the initial data may not bevery smooth and furthermore does not possess enough decay at space-like infinityto be in L2 Also we would like to understand how this can be done in situationswhere the equations being considered contain no special structure in the non-linearity For instance this is of interest in discussing the problem of small dataglobal well-posedness for the MaxwellndashKleinndashGordon and YangndashMills equationswith the Lorentz gauge enforced instead of the more regular Coulomb gauge Thisprovides a significant point of departure from earlier works on the global existencetheory of non-linear wave equations which for the general case requires precisecontrol on the initial data in certain weighted Sobolev spaces (see Klainerman 1985)or else requires the non-linearity to have some specific algebraic structure (perhapscoming from a gauge transformation) which allows one to exploit lsquolsquonull formrsquorsquoidentities or apply standard Strichartz estimates directly to the equation beingconsidered (see Tao 2001 Tataru 1998)

From the point of view of homogeneity we are lead directly to considerations ofthe low regularity properties of Eq (1) as follows By a simple scaling argumenta onecan see that the most efficient L2 based regularity assumption possible on the initialdata involves sc frac14 n

2 s derivatives Again by scale invariance and looking at unitfrequency initial datab one can see that if we are to impose only an L2 smallness con-dition on the initial data which contains no physical space weights then sc frac14 n

2 s isin fact the largest amount of derivatives we may work with This leads us to considerthe question of GWPS for initial data in the homogeneous Besov spaces _BBscp forvarious values of p In this work we will concentrate solely on the case p frac14 1 Thisis the strongest scale and translation invariant control on the initial data possibleand will be crucial for the kind of non-linearities we work with here In fact it doesnot seem possible to push any type of global regularity for equations of the type (1)which contain derivatives in the non-linearity down to the scale invariant Sobolevspace _HHsc frac14 _BBsc2 unless the equations under consideration possess a great deal ofspecial structure in the non-linearity This has been done for the wave-maps

aIn conjunction with finite time blowup for large data This phenomena is known to happenfor higher dimensional equations with derivative non-linearities even in the presence ofpositive conserved quantities (see eg Cazenave et al 1998)bThat is initial data sets where the Fourier transform is supported in the unit frequency

annulus fx 12 lt jxj lt 2g

Global Regularity for NLWE 1507

4_LPDE29_09amp10_R3_102804

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equations (see Tao 2001) and more recently for the MaxwellndashKleinndashGordonequations in eth6thorn 1THORN and higher dimensions with the help of the Coulomb gauge(see Rodnianski preprint) Both of these results depend crucially on the fact thatthe underlying gauge group of the equations is compact

In recent years there has been much progress in our understanding of the lowregularity local theory for general non-linear wave equations of the form (1) Inthe lower dimensional setting ie when n frac14 2 3 4 it is known from counterexam-ples of Lindblad (see Lindblad 1996) that there is ill posedness for initial data inthe Sobolev space Hs0 where s0 sc thorn 5n

4 Intimately connected with this phenom-ena is the failure of certain space-time estimates for the linear wave equation knownas Strichartz estimates Specifically one does not have anything close to an L2ethL4THORNestimate in these dimensions Such an estimate obviously plays a crucial role (viaDuhamelrsquos principle) in the quadratic theory However using the Strichartz esti-mates available in these dimensions along with Picard iteration in certain functionspaces one can show that the Lindblad counterexamples are sharp in that thereis local well-posedness for initial data in the spaces Hs when sc thorn 5n

4 lt s (see forexample Klainerman and Selberg 2002)

In the higher dimensional setting ie when the number of spatial dimensions isn frac14 5 or greater one does have access to Strichartz estimates at the level of L2ethL4THORN(see Keel and Tao 1998) and it is possible to push the local theory down to HscthornEwhere 0 lt E is arbitrary (see Tataru 1999)

In all dimensions the single most important factor which determines the localtheory as well as the range of validity for Strichartz estimates is the existence of freewaves which are highly concentrated along null directions in Minkowski spaceThese waves known as Knapp counterexamples resemble a single beam of lightwhich remains coherent for a long period of time before dispersing For a specialclass of non-linearities known as lsquolsquonull structuresrsquorsquo interactions between thesecoherent beams are effectively canceled and one gains an improvement in the localtheory of equations whose nonlinearities have this form (see for example Klainermanand Machedon 1993 Klainerman and Selberg 2002)

In both high and low dimensional settings the analysis of certain null structuresspecifically non-linearities containing the Q0 null fromc has led to the proof that thewavendashmaps model equationsd are well posed in the scale invariant lsquo1 Besov space _BB

n21

(see Tataru 1998 2001) While the proof of this result is quite simple for high dimen-sions it relies in an essential way on the structure of the Q0 null form In fact there isno direct way to extend the proof of this result to include the less regular nonlinea-rities of the form fHf or for that matter the Qij null formse which show up in theequations of gauge field theory However the high dimensional non-linear interac-tion of coherent waves is quite weak (eg giving the desired range of validity forStrichartz estimates) and one would expect that it is possible to prove local wellposedness for quadratic equations with initial data in the scale invariant lsquo1 Besov

cThis is defined by the equation Q0ethfcTHORN frac14 afacdNot the rough schematic we have listed here but rather equations of the formampf frac14 GethfTHORNQ0ethffTHORNeThese are defined by QijethfcTHORN frac14 ifjc jfic

1508 Sterbenz

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space without resorting to any additional structure in the nonlinearity For n frac14 5dimensions it may be that this is not quite possible although we provide no convin-cing evidence except for the fact that there is no obvious way to add over our loca-lized estimates in that dimension in order to obtain a full set of estimates that worksin all of spacendashtime Fourier space In fact every estimate we prove here leads to alogarithmic divergence in the distance to the cone in Fourier space for the case ofeth5thorn 1THORN dimensions so in this sense our argument breaks down completely in thatregime However for n frac14 6 and higher dimensions we will prove that in fact no nullstructure is needed for there to be well posedness in _BBsc1 This leads to the statementof our main result which is as follows

Theorem 11 (Global Well Posedness) Let 6 n be the number of spatial dimen-sions For any of the generic equations listed above YM WM or MD let ethf gTHORN be a(possibly vector valued) initial data set Let sc frac14 n

2 s be the corresponding L2

scaling exponent Then there exists constants 0 lt E0C such that if

kethf gTHORNk _BBsc 1 _BBsc11 E0 eth5THORN

there exits a global solution c which satisfies the continuity condition

kckCeth _BBsc 1THORNCeth1THORNeth _BBsc11THORN Ckethf gTHORNk _BBsc 1 _BBsc11 eth6THORN

The solution c is unique in the following sense There exists a sequence of smoothfunctions ethfN gN THORN such that

limN1

kethf gTHORN ethfN gN THORNk _BBsc 1 _BBsc11 frac14 0

For this sequence of functions there exists a sequence of unique smooth global solu-tions cN of (1) with this initial data Furthermore the cN converge to c as follows

limN1

kc cNkCeth _BBsc 1THORNCeth1THORNeth _BBsc11THORN frac14 0

Also c is the only solution which may be obtained as a limit (in the above sense) ofsolutions to (1) with regularizations of ethf gTHORN as initial data Finally c retains anyextra smoothness inherent in the initial data That is if ethf gTHORN also has finite_HHs _HHs1 norm for sc lt s then so does c at fixed time and one has the followingestimate

kckCeth _HHsTHORNCeth1THORNeth _HHs1THORN Ckethf gTHORNk _HHs _HHs1 eth7THORN

In a straightforward way the function spaces we iterate in allow us to show thefollowing scattering result

Theorem 12 Using the same notation as above we have that there exists data setsethf gTHORN such that if c is the solution to the homogeneous wave equation with the


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corresponding initial data the following asymptotics hold

limt1kcthorn ckCeth _BBsc 1THORNCeth1THORNeth _BBsc11THORN frac14 0 eth8THORN

limt1kc ckCeth _BBsc 1THORNCeth1THORNeth _BBsc11THORN frac14 0 eth9THORN

Furthermore the scattering operator retains any additional regularity inherent inthe initial data That is if ethf gTHORN has finite _HHs norm then so does ethf gTHORN andthe following asymptotics hold

limt1kcthorn ckCeth _HHsTHORNCeth1THORNeth _HHs1THORN frac14 0 eth10THORN

limt1kc ckCeth _HHsTHORNCeth1THORNeth _HHs1THORN frac14 0 eth11THORN

2 PRELIMINARY NOTATION

For quantities A and B we denote by AB to mean that A C B for somelarge constant C The constant C may change from line to line but will alwaysremain fixed for any given instance where this notation appears Likewise we usethe notation A B to mean that 1

C B A C B We also use the notation

A B to mean that A 1C B for some large constant C This is the notation we will

use throughout the paper to break down quantities into the standard cases A Bor A B or B A and AB or B A without ever discussing which constantswe are using

For a given function of two variables etht xTHORN 2 R R3 we write the spatial andspace-time Fourier transform as

ffetht xTHORN frac14Z

e2pixxfetht xTHORNdx

~ffetht xTHORN frac14Z

e2piethttthornxxTHORN fetht xTHORNdt dx

respectively At times we will also write Ffrac12f frac14 ~ff For a given set of functions of the spatial variable only we denote byWethf gTHORN the

solution of the homogeneous wave equation with Cauchy data ethf gTHORN If F is afunction on spacendashtime we will denote by WethFTHORN the function W Feth0THORN tFeth0THORNeth THORN

Let E denote any fundamental solution to the homogeneous wave equation ieone has the formula ampE frac14 d We define the standard Cauchy parametrix for thewave equation by the formula

amp1F frac14 E F WethE FTHORN

1510 Sterbenz

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Explicitly one has the identity

damp1Famp1Fetht xTHORN frac14 Z t

0

sineth2pjxjetht sTHORNTHORN2pjxj

bFFeths xTHORNds eth12THORN

For any function F which is supported away from the light cone in Fourierspace we shall use the following notation for division by the symbol of the waveequation

X1F frac14 E F

Of course the definition of X1 does not depend on E so long as F is supported awayfrom the light cone in Fourier space for us this will always be the case when thisnotation is in use Explicitly one has the formula

F X1F etht xTHORN frac14 1

4p2etht2 jxj2THORNeFFetht xTHORN

3 MULTIPLIERS AND FUNCTION SPACES

Let j be a smooth bump function (ie supported on the set jsj 2 such thatj frac14 1 for jsj 1) In what follows it will be a great convenience for us to assumethat j may change its exact form for two separate instances of the symbol j (evenif they occur on the same line) In this way we may assume without loss of generalitythat in addition to being smooth we also have the idempotence identity j2 frac14 j Weshall use this convention for all the cutoff functions we introduce in the sequel

For l 2 f2j j 2 Zg we denote the dyadic scaling of j by jlethsTHORN frac14 jethslTHORN Themost basic Fourier localizations we shall use here are with respect to the space-timevariable and the distance from the cone Accordingly for ld 2 f2j j 2 Zg we formthe LittlewoodndashPaley type cutoff functions

sletht xTHORN frac14 j2lethjetht xTHORNjTHORN j12lethjetht xTHORNjTHORN eth13THORN

cdetht xTHORN frac14 j2dethjtj jxjTHORN j12dethjtj jxjTHORN eth14THORN

We now denote the corresponding Fourier multiplier operator via the formulasfSluSlu frac14 sl~uu and gCduCdu frac14 cd~uu respectively We also use a multi-subscript notation todenote products of the above operators eg Sld frac14 SlCd We shall use the notation

Sld frac14Xdd

Sld eth15THORN

to denote cutoff in an OethdTHORN neighborhood of the light cone in Fourier space At timesit will also be convenient to write Sld frac14 Sl Slltd We shall also use the notationSld etc to denote the multiplier Sld cutoff in the half space t gt 0


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The other type of Fourier localization which will be central to our analysis is thedecomposition of the spatial variable into radially directed blocks of various sizesTo begin with we denote the spatial frequency cutoff by

plethxTHORN frac14 j2lethjxjTHORN j12lethjxjTHORN eth16THORN

with Pl the corresponding operator For a given parameter d l we now decom-pose Pl radially as follows First decompose the the unit sphere Sn1 Rn intoangular sectors of size d

l dl with bounded overlap (independent of d) These

angular sectors are then projected out to frequency l via rays through the originThe result is a decomposition of suppfplg into radially directed blocks of sizel d d with bounded overlap We enumerate these blocks and label thecorresponding partition of unity by bold It is clear that things may be arranged sothat upon rotation onto the x1-axis each bold satisfies the bound

jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN

In particular each Bold is given by convolution with an L1 kernel We shall also

denote

Sold frac14 Bo

lethldTHORN12Sld Sold frac14 Bo

lethldTHORN12Sld

Note that the operators Sold and Sold are only supported in the region wherejtj jxj

We now use these multipliers to define the following dyadic norms which will bethe building blocks for the function spaces we will use here

kukpX

12lp

frac14Xd22Z

dp

2kSldukpL2 (lsquolsquoclassicalrsquorsquo Hsd) eth18THORN

kukYl frac14 l1kampSlukL1ethL2THORN (Duhamel) eth19THORN

kukZlfrac14 l

2n2

Xd

Xo

kSolduk2L1ethL1THORN

12

(outer block) eth20THORN

Notice that the (semi) norms X12

lp and Yl are only well defined modulo measuressupported on the light cone in Fourier space Because of this it will be convenientfor us to include an extra L1ethL2THORN norm in the definition of our function spaces Thisrepresents the inclusion in the above norms of solutions to the wave equation withL2 initial data Adding everything together we are led to define the followingfixed frequency (semi) norms

Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN

Unfortunately the above norm is alone not strong enough for us to be able to iterateequations of the form (1) which contain derivatives This is due to a very specific

1512 Sterbenz

4_LPDE29_09amp10_R3

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Low High frequency interaction in quadratic non-linearities Fortunately thisproblem has been effectively handled by Tataru (1999) based on ideas fromKlainerman and Machedon (1996) and Klainerman and Tataru (1999) What isnecessary is to add some extra L1ethL1THORN norms on lsquolsquoouter blockrsquorsquo regions of Fourierspace This is the essence of the norm (20) above which is a slight variant of thatwhich appeared in Tataru (1999) This leads to our second main dyadic norm

Gl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN Zl eth22THORN

Finally the spaces we will iterate in are produced by adding the appropriate numberof derivatives combined with the necessary Besov structures

kuk2Fs frac14Xl

l2skuk2Fl eth23THORN

kukGs frac14Xl

lskukGl eth24THORN

Due to the need for precise microlocal decompositions of crucial importance tous will be the boundedness of certain multipliers on the components (18)ndash(19) of ourfunction spaces as well as mixed Lebesgue spaces We state these as follows

Lemma 31 (Multiplier Boundedness)

(1) The following multipliers are given by L1 kernels l1HSl Sold Sold and

ethldTHORNX1Sold In particular all of these are bounded on every mixedLebesgue space LqethLrTHORN

(2) The following multipliers are bounded on the spaces LqethL2THORN for1 q 1 Sld and Sld

Proof of Lemma 31 (1) First notice that after a rescaling the symbol for themultiplier l1HSl is a C1 bump function with Oeth1THORN support Thus its kernel is inL1 with norm independent of l

For the remainder of the operators listed in (1) above it suffices to work withethldTHORNX1Sold The boundedness of the others follows from a similar argument Welet w denote the symbol of this operator cut off in the upper resp lower half planeAfter a rotation in the spatial domain we may assume that the spatial projection ofw is directed along the positive x1 axis Now look at wthorneths ZTHORN with coordinates

s frac14 1ffiffiffi2

p etht x1THORN

Z1 frac14 1ffiffiffi2

p ethtthorn x1THORN

Z0 frac14 x0

It is apparent that wthorneths ZTHORN has support in a box of dimension lffiffiffiffiffiffild

p ffiffiffiffiffiffild

p d with sides parallel to the coordinate axis and longest side in


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the Z1 direction and shortest side in the s direction Furthermore a directioncalculation shows that one has the bounds

jNZ1wthornj CNl

N jNZ0 w

thornj CN ethldTHORNN=2 jNs w

thornj CNdN

Therefore we have that wthorn yields an L1 kernel A similar argument works for thecutoff function w using the rotation


p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp

Proof of Lemma 31 eth2THORN We will argue here for Sld The estimates for the othersfollow similarly If we denote by Ketht xTHORN the convolution kernel associated with Sldthen a simple calculation shows that

e2pitjxjcKKetht xTHORN frac14Z

e2pittcetht xTHORNdt

where suppfcg is contained in a box of dimension l l d with sides alongthe coordinate axis and short side in the t direction Furthermore one has theestimate

jNt cj CNd

N

This shows that we have the bound

kcKKkL1t ethL1

x THORN 1

independent of l and d Thus we get the desired bounds for the convolutionkernels amp

As an immediate application of the above lemma we show that the extra Zl

intersection in the Gl norm above only effects the X12

l1 portion of things

Lemma 32 (Outer Block Estimate on Yl) For 5 lt n one has the following uniforminclusion

Yl 13 Zl eth25THORN

1514 Sterbenz

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Proof of (25) It is enough to show that

Xo

kX1Solduk2L1ethL1THORN

12

ln42

d

l

n54

kSlukL1ethL2THORN

First using a local Sobolev embedding we see that

kBo

lethldTHORN12X1SoldukL1ethL1THORN l

nthorn14 d

n14 kX1SoldukL1ethL2THORN

Therefore using the boundedness Lemma 31 it suffices to note that by Minkowskirsquosinequality we can bound

Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o

kSolduethtTHORNk2L2x

12

dt

kSldukL1ethL2THORN

The last line of the above proof showed that it is possible to bound a squaresum over an angular decomposition of a given function in L1ethL2THORN It is also clear that

this same procedure works for the X12

l1 spaces because one can use Minkowskirsquosinequality for the lsquo1 sum with respect to the cone variable d This fact will be of greatimportance in what follows and we record it here as

Lemma 33 (Angular Reconstruction of Norms) Given a test function u andparameter d l one can bound

Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN

4 STRUCTURE OF THE Fk SPACES

The purpose of this section is to clarify some remarks of the previous section andwrite down two integral formulas for functions in the Fl space This material is allmore or less standard in the literature (see eg Foschi and Klainerman 2000 Tataru1998 1999) and we include it here primarily because the notation will be useful forour scattering result Our first order of business is to write down a decomposition forfunctions in the Fl space

Lemma 41 (Fl Decomposition) For any ul 2 Fl one can write

ul frac14 uXlthorn u

X1=2

l1thorn uYl eth27THORN


4_LPDE29_09amp10_R3_102804

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where uXlis a solution to the homogeneous wave equation u

X1=2

l1is the Fourier

transform of an L1 function and uYl satisfies

uYleth0THORN frac14 tuYleth0THORN frac14 0

Furthermore one has the norm bounds

1

CkulkFl

ku

XlkL1ethL2THORN thorn ku

X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN

We now show that the two inhomogeneous terms on the right hand side of (27)can be written as integrals over solutions to the wave equation with L2 data Thisfact will be of crucial importance to us in the sequel The first formula is simply arestatement of (12)

Lemma 42 (Duhamelrsquos Principle) Using the same notation as above for any uYl one can write

uYlethtTHORN frac14 Z t

0

jDxj1 sinetht sTHORNjDxj

ampuYlethsTHORNds eth29THORN

Likewise one can write the uX

1=2

l1portion of the sum (27) as an integral over

modulated solutions to the wave equation be foliating Fourier space by forwardand backward facing light-cones

Lemma 43 (X12

l1 Trace Lemma) For any uX

1=2

l1 let u

X1=2

l1

denote its restriction to the

frequency half space 0 lt t Then one can write

uX

1=2

l1

ethtTHORN frac14Z

e2pitseitjDxjuls ds eth30THORN

where uls is the spatial Fourier transform of fuu restricted to the sth translate of theforward or backward light-cone light cone in Fourier space ie

cuulsethxTHORN frac14Z

detht s jxjTHORNfuuetht xTHORNdtIn particular one has the formulaZ

kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN

5 STRICHARTZ ESTIMATES

Our inductive estimates will be based on a method of bilinear decompositionsand local Strichartz estimates as in the work Tataru (1999) We first state thestandard Strichartz from which the local estimates follow

1516 Sterbenz

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Lemma 51 (Homogeneous Strichartz Estimates see Keel and Tao 1998) Let5 lt n s frac14 n1

2 and suppose u is a given function of the spatial variable only Thenif 1

qthorn s

r s

2 and1qthorn n

rfrac14 n

2 g the following estimate holds

keitjDxjPlukLqt ethLr

xTHORN lgkPlukL2 eth32THORN

Combining the L2ethL2ethn1THORNn3 THORN endpoint of the above estimate with a local Sobolev in the

spatial domain we arrive at the following local version of (32)

Lemma 52 (Local Strichartz Estimate) Let 5 lt n then the following estimateholds

keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo

lethldTHORN12ukL2 eth33THORN

Using the integral formulas (29) and (30) we can transfer the above estimates tothe Fl spaces

Lemma 53 (Fl Strichartz Estimates) Let 5 lt n and set s frac14 n12 Then if 1

qthorn s

r s

2

and 1qthorn n

rfrac14 n

2 g the following estimates hold

kSlukLqethLrTHORN lgkukFl eth34THORN

Xo


12

lnthorn14 d

n34 kukFl

eth35THORN

Proof of Lemma 53 It suffices to prove the estimate (34) as the estimate (35)follows from this and a local Sobolev embedding combined with the re-summingformula (26) Using the decomposition (41) and the angular reconstruction formula(26) it is enough to prove (34) for functions u

X1=2

l1and uYl Using the integral formula

(30) we see immediately that

kuX

1=2

l1kLqethLrTHORN

X

ZkeitjDxjulskLqethLrTHORN ds

lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1

For the uYl portion of things we can chop the function up into a fixed number ofspacendashtime angular sectors using L1 convolution kernels Doing this and using Ra


4_LPDE29_09amp10_R3_102804

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to denote an operator from the set fI ijDxj1g we estimate

kuYlkLqethLrTHORN l1Xa

kauYlkLqethLrTHORN

l1Xa

ZkeitjDxjeisjDxjRaampuYleths xTHORN

kLqt ethLr

xTHORN ds

lgl1Xa

ZkeisjDxjRaampuYleths xTHORNkL2

xds

lg kuYlkYl amp

A consequence of (34) is that we have the embedding

X12

l1 13 L1ethL2THORN

Using a simple approximation argument along with uniform convergence we arriveat the following energy estimate for the Fs and Gs spaces

Lemma 54 (Energy Estimates) For space-time functions u one has the followingestimates

kukCeth _HHsTHORNCeth1THORNeth _HHs1THORN kukFs eth36THORN

kukCeth _BBsTHORNCeth1THORNeth _BBs1THORN kukGs eth37THORN

Also by duality and the estimate (5) we have that

lX1L1ethL2THORN 13 lN1X1

2

l1 13 X12

l1 eth38THORN

This proves shows

Lemma 55 (L2 Estimate for Yl) The following inclusion holds uniformly

d12SldethYlTHORN 13 L2ethL2THORN eth39THORN

in particular by dyadic summing one has

d12SldethFlTHORN 13 L2ethL2THORN

6 SCATTERING

It turns out that our scattering result Theorem 12 is implicitly contained in thefunction spaces Fs and Gs That is there is scattering in these spaces independentlyof any specific equation being considered Therefore to prove Theorem 12 it willonly be necessary to show that our solution to (1) belongs to these spaces

1518 Sterbenz

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Using a simple approximation argument it suffices to deal with things at fixedfrequency In what follows we will denote the space _HH1 1

t ethL2THORN to be the Banachspace with fixed time energy norm

kuethtTHORNk2_HH11t ethL2THORN frac14 kuethtTHORNk2_HH1 thorn ktuethtTHORNk2L2

Because the estimates in Theorem 12 deal with more than one derivative we willshow that

Lemma 61 (Fl Scattering) For any function ul 2 Fl there exists a set of initialdata ethf

l gl THORN 2 PlethL2THORN lPlethL2THORN such that the following asymptotic holds

limt1kulethtTHORN Wethfthorn

l gthornl THORNethtTHORNk _HH11

t ethL2THORN frac14 0 eth40THORN

limt1kulethtTHORN Wethf

l gl THORNethtTHORNk _HH11


Proof of Lemma 61 Using the notation of Sec 4 we may write

ul frac14 uXlthorn uthorn

X1=2

l1

thorn uX

1=2

l1

thorn uYl

We now define the scattering data implicitly by the relations

Wethfthornl g

thornl THORNethtTHORN frac14 u

XlZ 1

0

jDxj1 sin jDxjetht sTHORNeth THORNampuYlethsTHORNds

Wethfl g

l THORNethtTHORN frac14 u

XlZ 0

1jDxj1 sin jDxjetht sTHORNeth THORNampuYlethsTHORNds

Using the fact that ampuYl has finite L1ethL2THORN norm it suffices to show that one has the

limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1

ethtTHORNk _HH11t ethL2THORN frac14 0

Squaring this we see that we must show the limits

limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1

ethtTHORN frac14 0 eth42THORN

limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1


Wersquoll only deal here with the limit (42) as the limit (43) follows from a virtuallyidentical argument Using the trace formula (30) along with the Plancherel theorem


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we compute

ethLHSTHORNeth42THORNfrac14 lim

t14p2

Ze2pitethjxjjxjTHORNjxj2

Ze2pits1 duthornls1thorns2

uthornls1thorns2ethxTHORN duls2uls2ethxTHORNds1 ds2 dx

By (31) we have the bounds

jxj2Z duthornls1thorns2

uthornls1thorns2ethxTHORN duls2uls2ethxTHORN

ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1

This shows that the function

HtethxTHORN frac14 jxj2Z

e2pits1 duthornls1thorns2uthornls1thorns2

ethxTHORN duls2uls2ethxTHORN ds1 ds2

is bounded point-wise by an L1 function uniformly in t Therefore by the dominatedconvergence theorem it suffices to show that we in fact have that limt1 Ht frac14 0To see this notice that by the above bounds in conjunction with Fubinirsquos theoremwe have that for almost every fixed x the integral


uthornls1thorns2ethxTHORN duls2uls2ethxTHORN ds2

is in L1s1 The result now follows from the Riemann Lebesgue Lemma Explicitly one

has that for almost every fixed x the following limit holds

limt1

HtethxTHORN frac14 limt1

jxj2Z


ethxTHORN duls2uls2ethxTHORNds1 ds2 frac14 0 amp

7 INDUCTIVE ESTIMATES I

Our solution to (1) will be produced through the usual procedure of Picard itera-tion Because the initial data and our function spaces are both invariant with respectto the scaling (3) any iteration procedure must effectively be global in time There-fore we shall have no need of an auxiliary time cutoff system as in the worksKlainerman and Machedon (1995) and Tataru (1999) Instead we write (1) directlyas an integral equation

f frac14 Wethf gTHORN thornamp1NethfDfTHORN eth44THORN

By the contraction mapping principle and the quadratic nature of the nonlinearityto produce a solution to (44) which satisfies the regularity assumptions of our maintheorem it suffices to prove the following two sets of estimates

Theorem 71 (Solution of the Division Problem) Let 5 lt n then the F and G

spaces solve the division problem for quadratic wave equations in the sense that

1520 Sterbenz

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for any of the model systems we have written above YM WM or MD one has thefollowing estimates

kamp1NethuDvTHORNkGsc kukGsckvkGsc eth45THORN

kamp1NethuDvTHORNkFs kukGsc kvkFs thorn kukFskvkGsc eth46THORN

The remainder of the paper is devoted to the proof of Theorem 71 In whatfollows we will work exclusively with the equation

f frac14 Wethf gTHORN thornamp1ethfHfTHORN eth47THORN

In this case we set sc frac14 n22 The proof of Theorem 71 for the other model equations

can be achieved through a straightforward adaptation of the estimates we give hereIn fact after the various derivatives and values of sc are taken into account the proofin these cases follows verbatim from estimates (49) and (50) below

Our first step is to take a LittlewoodndashPaley decomposition of amp1ethuHvTHORN withrespect to space-time frequencies

amp1ethuHvTHORN frac14Xmi

amp1ethSm1uHSm2vTHORN eth48THORN

We now follow the standard procedure of splitting the sum (48) into three piecesdepending on the cases m1 m2 m2 m1 and m2 m1 Therefore due to the lsquo1 Besovstructure in the F spaces in order to prove both (45) and (46) it suffices to show thetwo estimates

kamp1ethSm1uHSm2vTHORNkGl l1m

n2

1kukFm1kvkFm2

m1 m2 eth49THORN

kamp1ethSmuHSlvTHORNkGl m

n22 kukGm

kvkFl m l eth50THORN

Notice that after some weight trading the estimates (45) and (46) follow from (50) inthe case where m2m1

Proof of (49) It is enough if we show the following two estimates

kSlethSm1uHSm2vTHORNkL1ethL2THORN mn2

1kukFm1kvkFm2

m1 m2 eth51THORN

kSlamp1ethSm1uHSm2vTHORNkL1ethL2THORN l1mn2

1kukFm1kvkFm2

m1 m2 eth52THORN

In fact it suffices to prove (51) To see this notice that one has the formula

Slamp1

G frac14 WethE SlGTHORN SlWethE GTHORN


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Thus after multiplying by Sl we see that

Sl Slamp1

G frac14 PlWethE SlGTHORN SlWethE GTHORN

frac14 WethE SlPlGTHORN SlWethE PlGTHORN

frac14 Sl Slamp1

PlG

Therefore by the (approximate) idempotence of Sl one has

Slamp1G frac14 Slamp1SlGthorn Sl Slamp1

G

frac14 Slamp1SlGthorn Sl Slamp1

PlG

Thus by the boundedness of Sl on the spaces L1ethL2THORN the energy estimate one canbound

kSlamp1GkL1ethL2THORN l1 kSlGkL1ethL2THORN thorn kPlGkL1ethL2THORN

We now use the fact that the multipliers Sl and Pl are both bounded on the spaceL1ethL2THORN to reduce things to the estimate

kSm1uHSm2vkL1ethL2THORN kSm1ukL2ethL4THORNkHSm2vkL2ethL4THORN

mn24

1 mnthorn24

2 kukFm1kvkFm2

Taking into account the bound m1 m2 the claim now follows amp

Next wersquoll deal with the estimate (50) For the remainder of the paper we shallfix both l and m and assume they such that m l for a fixed constant We nowdecompose the product SlethSmuHSlvTHORN into a sum of three pieces

SlethSmuHSlvTHORN frac14 Athorn Bthorn C eth53THORN

where

A frac14 SlethSmuHSlcmvTHORNB frac14 SlcmethSmuHSlltcmvTHORNC frac14 SlltcmethSmuHSlltcmvTHORN

Here c is a suitably small constant which will be chosen later It will be needed tomake explicit a dependency between some of the constants which arise in a specificfrequency localization in the sequel We now work to recover the estimate (50) foreach of the three above terms separately

1522 Sterbenz

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Proof of (50) for the Term A Following the remarks at the beginning of the proofof (49) it suffices to compute

kSlethSmuHSlcmvTHORNkL1ethL2THORN l kSmukL2ethL1THORNkSlcmvkL2ethL2THORN

lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl

For a fixed c we obtain the desired result amp

We now move on to showing the inclusion (50) for the B term above In thisrange we are forced to work outside the context of L1ethL2THORN estimates This is thereason we have included the L2ethL2THORN based X

12

l1 spaces This also means that we willneed to recover Zl norms by hand (because they are only covered by the Yl spaces)However because this last task will require a somewhat finer analysis than what wewill do in this section we content ourselves here with showing

Proof of the X12

l1 SlethL1ethL2THORNTHORN Estimates for the Term B Our first task will be dealwith the energy estimate which we write as

kSlamp1SlcmethSmuHSlltcmvTHORNkL1ethL2THORNmn22 kukFm

kvkFl eth54THORN

For G supported away from the light-cone in Fourier space we have the identity

Slamp1SlG frac14 X1SlGWethX1PlSlGTHORN

Therefore by using the energy estimate for the X12

l1 space this allows us to estimate

kSlamp1SlGkL1ethL2THORN kX1SlGkL1ethL2THORN thorn kWethX1PlSlGTHORNkL1ethL2THORN

kX1SlGkL1ethL2THORN

kX1SlGkX

12l1

Therefore in order to prove the estimate (54) we are left with estimating the term B

in the X12

l1 space To do this for a fixed distance d from the cone we compute that

kX1SldSlcmethSmuHSlltcmvTHORNkL2ethL2THORN d1 kSmukL2ethL1THORNkSlvkL1ethL2THORN

d1mn22 kvkFm

kukFl


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Summing d12 times this last expression over all cm d yieldsX

cmd

d12kX1SldSlcmethSmuHSlltcmvTHORNkL2ethL2THORN

Xcmd

md

12

mn22 kvkFm

kukFl


8 INTERLUDE SOME BILINEAR DECOMPOSITIONS

To proceed further it will be necessary for us to take a closer look at theexpression

SoldethSmuHSlltcmvTHORN cm d eth55THORN

as well as the C term from line (53) above which we write as the sum

C frac14 SlltcmethSmuHSlltcmvTHORN frac14 CI thorn CII thorn CIII

where

CI frac14Xdltcm

SldethSmdu HSldvTHORN

CII frac14Xdltcm

SlltdethSmdu HSldvTHORN

CIII frac14Xdm

SlltminfcmdgethSmdu HSlltminfcmdgvTHORN

Wersquoll begin with a decomposition of CI and CII The CIII term is basically thesame but requires a slightly more delicate analysis All of the decompositions wecompute here will be for a fixed d The full decomposition will then be given by sum-ming over the relevant values of d Because our decompositions will be with respectto Fourier supports it suffices to look at the convolution product of the correspond-ing cutoff functions in Fourier space In what follows wersquoll only deal with the CI

term It will become apparent that the same idea works for CII Therefore withoutloss of generality we shall decompose the product

sthornld smd sthornld

eth56THORN

To do this we use the standard device of restricting the angle of interaction inthe above product It will be crucial for us to be able to make these restrictionsbased only on the spatial Fourier variables because we will need to reconstructour decompositions through square-summing For etht0 x0THORN 2 suppfsmdg and

1524 Sterbenz

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etht xTHORN 2 suppfsthornldg we compute that

OethdTHORN frac14 jt0 thorn tj jx0 thorn xjfrac14 jx0j thorn jxj thorn OethdTHORN jx0 thorn xj

frac14OethdTHORN thorn jx0j thorn jxj jx0 thorn xj

Using now the fact that d lt cm and m lt cl to conclude that jx0j m and jxj l wesee that one has the angular restriction

mY2x0x

jx0j thorn jxj jx0 thorn xjfrac14OethdTHORN

In particular we have that Yx0x ffiffidm

q This allows us to decompose the product (56)

into a sum over angular regions with O

ffiffidm

q spread The result is

Lemma 81 (Wide Angle Decomposition) In the ranges stated for the CI termabove one can write

sthornldethsmd sthornldTHORN frac14X

o1 o2 o3

jo1o2 jethd=mTHORN12


bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN

for the convolution of the associated cutoff functions in Fourier space

We note here that the key feature in the decomposition (57) is that the sum is(essentially) diagonal in all three angles which appear there (o1o2o3) It is usefulto keep in mind the diagram (Fig 1)

Figure 1 Spatial supports in the wide angle decomposition


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We now focus our attention on decomposing the convolution

sthornlminfcmdgsmd sthornlminfcmdg

eth58THORN

If it is the case that d m then the same calculation which was used to produce (57)works and we end up with the same type of sum However if we are in the case whered m we need to compute things a bit more carefully in order to ensure that we maystill decompose the multiplier smd using only restrictions in the spatial variable Todo this we will now assume that things are set up so that cm d It is clear thatall the previous decompositions can be made so that we can reduce things to thisconsideration If we now take etht0 x0THORN 2 suppfsmdg and etht xTHORN 2 suppfsthornlminfcmdggwe can use the facts that t0 frac14 OethdTHORN jx0j t frac14 OethcmTHORN thorn jxj and jxj m to computethat

OethcmTHORN frac14t0 thorn t

jx0 thorn xj

frac14OethdTHORN jx0j thorn OethcmTHORN thorn jxj jx0 thorn xj

eth59THORN

where the term OethdTHORN in the above expression is such that jOethdTHORNj d In fact onecan see that the equality (59) forces OethdTHORN lt 0 on account of the fact thatethjx0j thorn jxj jx0 thorn xjTHORN gt 0 and the assumption jOethcmTHORN thorn OethdTHORNj d In particularthis means that we can multiply smd in the product (58) by the cutoff sjtjltjxj withouteffecting things This in turn shows that we may decompose the product (58) basedsolely on restriction of the spatial Fourier variables just as we did to get the sum inLemma 81

We now return to the CI term For the sequel we will need to know whatthe contribution of the factor Sldv to the following localized product is

So1

ldethSmdu HSldvTHORN

Using Lemma 81 we see that we may write

so1

ldethsmd sldTHORN frac14 so1

ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN

where jo1 o2j jo3 o2j ffiffidm

q However this can be refined significantly To

see this assume that the spatial support of so1

ld lies along the positive x1 axis Wersquolllabel this block by b

o1

lethldTHORN12 Because we are in the range where

ffiffiffiffiffiffimd


p and

furthermore because for every x 2 suppfbo3

llethdmTHORN12

g and x0 2 suppx0 fso2

mdg the sum

xthorn x0 must belong to suppfbo1

lethldTHORN12g we in fact have that x itself must belong to a

1526 Sterbenz

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block of size l ffiffiffiffiffiffild


p This allows us to write

Lemma 82 (Small Angle Decomposition) In the ranges stated for the CI termabove we can write

so1


ld

so2

md so3

ld

eth61THORN

where jo1 o3j ffiffidl

q and jo1 o2j

ffiffidm

q

It is important to note here that if one were to sum the expression (60) over o1the resulting sum would be (essentially) diagonal in o3 but there would be many o1

which would contribute to a single o2 This means that the resulting sum would notbe diagonal in o2 as was the case for the sum (57) It is helpful to visualize thingsthrough the Fig 2

Our final task here is to mention an analog of Lemma 82 for the term (55) Herewe can frequency localize the factor Slltcm in the product using the fact that one hasm c

12

ffiffiffiffiffiffild

p The result is

Lemma 83 (Small Angle Decomposition for the Term B) In the ranges stated forthe B term above we can write

so1

ldethsm slcmTHORN frac14 so1

ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q

Finally we note here the important fact that in the decomposition (62) abovethe range of interaction in the product forces d m This completes our list ofbilinear decompositions

Figure 2 Spatial supports in the small angular decomposition


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9 INDUCTIVE ESTIMATES II REMAINDER OF THELowHigh)High FREQUENCY INTERACTION

It remains for us is to bound the term B from line (55) in the Zl space as well asshow the inclusion (50) for the terms CI ndash CIII from line (8) We do this now proceed-ing in reverse order

Proof of Estimate (50) for the CIII Term To begin with we fix d Using the remarksat the beginning of the proof of (49) we see that it is enough to show that

kSlltminfcmdgethSmdu HSlltminfcmdgvTHORNkL1ethL2THORN lX

o

kSomduk2L1ethL1THORN

12

kvkFl

eth63THORN

To accomplish this we first use the wide angle decomposition (57) on the left handside of (63) This allows us to compute using a CauchyndashSchwartz that

kSlltminfcmdgethSmdu HSlltminfcmdgvTHORNkL1ethL2THORN

Xo2 o3


kSo2

mdukL1ethL1THORN kHBo3

llethdmTHORN12

SlltminfcmdgvTHORNkL1ethL2THORN

lXo


12 X

o

kBo

llethdmTHORN12SlltminfcmdgvTHORNk2L1ethL2

xTHORN

12

lXo


12

kvkFl

Summing over d now yields the desired estimate amp

Proof of (50) for the CII Term Again fixing d and using the angular decomposi-tion Lemma 81 we compute that

kSlltdethSmdu HSldvTHORNkL1ethL2THORN

lXo2 o3


kSo2

mdukL2ethL1THORN kBo3

llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS

This last expression can now be summed over d using the condition d lt cm toobtain the desired result amp

Proof of (50) for the CI Term This is the other instance where we will have to relyon the X

12

l1 space Following the same reasoning used previously we first bound

kX1SldethSmdu HSldvTHORNkL2ethL2THORN

d1Xo2 o3


kSo2


llethdmTHORN12

SldvTHORNkL1ethL2THORN

d1Xo


12

Xo

kBo

llethdmTHORN12SldvTHORNk2L1ethL2

xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl

Multiplying this last expression by d12 and then using the condition d lt cm to sum

over d yields the desired result for the X12

l1 space part of estimate (50) It remainsto prove the Zl estimate Here we use the second angular decomposition Lemma 82to compute that for fixed d

Xo1

kX1So1

ldethSmdu HSldvTHORNk2L1ethL1THORN

12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo

kSomdukL2ethL1THORN Xo

kSoldvk2L2ethL1THORN

12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl

Multiplying this last expression by leth2nTHORNeth2THORN and summing over d using the condition

d lt l m yields the desired result amp

Proof of the Zl Embedding for the B Term The pattern here follows that of thelast few lines of the previous proof Fixing d we use the decomposition Lemma 83


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to compute that

Xo

kN1SoldethSmu HSlcmvTHORNk2L1ethL1THORN

12

ethldTHORN1Xo1 o3

jo1o3 jethd=lTHORN12

kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12

d1kSmukL2ethL1THORN Xo

kBo

lethldTHORN12Slcmvk2L2ethL1THORN

12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl

Multiplying the last line above by a factor of leth2nTHORNeth2THORN and using the conditions d lt l

and cm lt d m we may sum over d to yield the desired result amp

ACKNOWLEDGMENT

This work was conducted under NSF grant DMS-0100406

REFERENCES

Bournaveas N (1996) Local existence for the MaxwellndashDirac equations in threespace dimensions Comm Partial Differential Equations 21(5ndash6)693ndash720

Cazenave T Shatah J Shadi Tahvildar-Zadeh A (1998) Harmonic maps of thehyperbolic space and development of singularities in wave maps andYangndashMills fields Ann Inst H Poincare Phys Theor 68(3)315ndash349

Foschi D Klainerman S (2000) Bilinear space-time estimates for homogeneouswave equations Ann Sci cole Norm Sup (4) 33(2)211ndash274

Keel M Tao T (1998) Endpoint Strichartz estimates Am J Math 120(5)955ndash980

Klainerman S (1985) Uniform decay estimates and the Lorentz invariance of theclassical wave equation Comm Pure Appl Math 38(3)321ndash332

Klainerman S Machedon M (1993) Space-time estimates for null forms and thelocal existence theorem Comm Pure Appl Math 46(9)1221ndash1268

Klainerman S Machedon M (1995) Smoothing estimates for null forms andapplications Duke Math J 81(1)99ndash103

Klainerman S Machedon M (1996) Estimates for null forms and the spaces HsdInt Math Res Notices 17853ndash865

1530 Sterbenz

ORDER REPRINTS

Klainerman S Tataru D (1999) On the optimal local regularity for YangndashMillsequations in R4thorn1 J Am Math Soc 12(1)93ndash116

Klainerman S Selberg S (2002) Bilinear estimates and applications to nonlinearwave equations Commun Contemp Math 4(2)223ndash295

Lindblad H (1996) Counterexamples to local existence for semi-linear waveequations Am J Math 118(1)1ndash16

Rodnianski I Tao T Global regularity for the MaxwellndashKleinndashGordon equationin high dimensions Preprint

Tao T (2001) Global regularity of wave maps I Small critical Sobolev norm in highdimension Int Math Res Notices 6299ndash328

Tataru D (1998) Local and global results for wave maps I Comm PartialDifferential Equations 23(9ndash10)1781ndash1793

Tataru D (2001) On global existence and scattering for the wave maps equationAm J Math 123(1)37ndash77

Tataru D (1999) On the equationampu frac14 jHuj2 in 5thorn 1 dimensionsMath Res Lett6(5ndash6)469ndash485

Received October 2003Accepted January 2004


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1 INTRODUCTION

In this paper our aim is to give a more or less complete description of the globalregularity properties of generic homogeneous quadratic semi-linear wave equationson eth6thorn 1THORN and higher dimensional Minkowski space The equations we will considerare all of the form

ampf frac14 NethfDfTHORN eth1THORN

Here amp frac14 2t thorn Dx denotes the standard wave operator on Rnthorn1 and N is a

smooth function of f and its first partial derivatives which we denote by Df Forall of the nonlinearities we study here N will be assumed to be at least quadraticin nature that is

NethX Y THORN frac14 OethjethX Y THORNj2THORN ethX Y THORN 0

The homogeneity condition we require N to satisfy is that there exist a (vector) ssuch that

Nethlsf lsthorn1DfTHORN frac14 lsthorn2NethfDfTHORN eth2THORN

where we use multiindex notation for vector N The condition (2) implies thatsolutions to the system (1) are invariant (again solutions) if one performs the scaletransformations

fethTHORNff lsfethlTHORN eth3THORN

The general class of equations which falls under this description contains virtually allmassless non-linear field theories on Minkowski space including the Yang Millsequations (YM) the wavendashmaps equations (WM) and the MaxwellndashDirac equations(MD) We list the schematics for these systems respectively as

ampA frac14 ADAthorn A3 ethYMTHORNampf frac14 jDfj2 ethWMTHORNampu frac14 ADu ethMDTHORNampA frac14 jDuj2

The various values of s for these equations are (respectively) s frac14 1 s frac14 0 ands frac14 eth12 1THORN For a more complete introduction to these equations see for instancethe works Foschi and Klainerman (2000) and Bournaveas (1996) For the purposesof this paper will will only be concerned with the structure of these equations at thelevel of the generic schematics (YM)ndash(MD)

The central problem we will be concerned with is that of giving a precisedescription of the regularity assumptions needed in order to guarantee that the

1506 Sterbenz

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4_LPDE29_09amp10_R3_102804

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n21



1508 Sterbenz

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limN1



limN1







ORDER REPRINTS





















1510 Sterbenz

ORDER REPRINTS



0




X1F frac14 E F










Sld frac14Xdd

Sld eth15THORN



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS











4_LPDE29_09amp10_R3_102804

ORDER REPRINTS








n21



1508 Sterbenz

ORDER REPRINTS









limN1



limN1







ORDER REPRINTS





















1510 Sterbenz

ORDER REPRINTS



0




X1F frac14 E F










Sld frac14Xdd

Sld eth15THORN



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS








n21



1508 Sterbenz

ORDER REPRINTS









limN1



limN1







ORDER REPRINTS





















1510 Sterbenz

ORDER REPRINTS



0




X1F frac14 E F










Sld frac14Xdd

Sld eth15THORN



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS









limN1



limN1







ORDER REPRINTS





















1510 Sterbenz

ORDER REPRINTS



0




X1F frac14 E F










Sld frac14Xdd

Sld eth15THORN



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS





















1510 Sterbenz

ORDER REPRINTS



0




X1F frac14 E F










Sld frac14Xdd

Sld eth15THORN



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



0




X1F frac14 E F










Sld frac14Xdd

Sld eth15THORN



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS






jN1 b

oldj CNl

N jNi b

oldj CNd

N eth17THORN


denote

Sold frac14 Bo


lethldTHORN12Sld



kukpX

12lp

frac14Xd22Z

dp



kukZlfrac14 l

2n2

Xd

Xo


12




Fl frac14 X12

l1 thorn Yl

Sl L1ethL2THORN

eth21THORN


1512 Sterbenz

4_LPDE29_09amp10_R3

ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS


Gl frac14 X12

l1 thorn Yl



kuk2Fs frac14Xl


kukGs frac14Xl

lskukGl eth24THORN









p etht x1THORN


p ethtthorn x1THORN

Z0 frac14 x0





ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS


jNZ1wthornj CNl

N jNZ0 w


thornj CNdN



p ethtthorn x1THORN


p ethtthorn x1THORN

Z0 frac14 x0 amp





jNt cj CNd

N


kcKKkL1t ethL1

x THORN 1






Yl 13 Zl eth25THORN

1514 Sterbenz

ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS


Xo


12

ln42

d

l

n54

kSlukL1ethL2THORN


kBo


nthorn14 d



Xo

ZkSolduethtTHORNkL2

x

2

dt

0 1A12

Z X

o


12

dt

kSldukL1ethL2THORN





Xo

kBolduk2

X12l1Yl

12

kukX

12l1Yl

eth26THORN





X1=2



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS


X1=2

l1is the Fourier




1

CkulkFl

ku


X1=2

l1kX

12l1

thorn kuYlkYl

CkulkFl eth28THORN




0




1=2



Lemma 43 (X12


1=2

l1 let u

X1=2

l1



uX

1=2

l1

ethtTHORN frac14Z





kulskL2ds kuX

1=2

l1

kX

12l1

eth31THORN



1516 Sterbenz

ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



qthorn s

r s

2 and1qthorn n

rfrac14 n







keitjDxjBo

lethldTHORN12ukL2

t ethL1x THORN l

nthorn14 d

n34 kBo




qthorn s

r s

2

and 1qthorn n

rfrac14 n



Xo


12

lnthorn14 d

n34 kukFl

eth35THORN


X1=2



kuX

1=2

l1kLqethLrTHORN

X


lgX

ZkulskL2 ds

lgkuX

1=2

l1kX

12l1



4_LPDE29_09amp10_R3_102804

ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



kauYlkLqethLrTHORN

l1Xa


kLqt ethLr

xTHORN ds

lgl1Xa


xds

lg kuYlkYl amp


X12

l1 13 L1ethL2THORN







2

l1 13 X12

l1 eth38THORN

This proves shows





6 SCATTERING


1518 Sterbenz

ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS















X1=2

l1

thorn uX

1=2

l1

thorn uYl


Wethfthornl g


XlZ 1

0


Wethfl g


XlZ 0



limits

limt1

kuthornX

1=2

l1

ethtTHORN thorn uX

1=2

l1



limt1

ZjDxjuthorn

X1=2

l1

ethtTHORN jDxjuX

1=2

l1


limt1

Ztu

thornX

1=2

l1

ethtTHORN tuX

1=2

l1




ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS

we compute


t14p2







ds1 ds2 L1x

l2 kuthornX

1=2

l1

kX

12l1

kuX

1=2

l1

kX

12l1










limt1


jxj2Z









1520 Sterbenz

ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS













n2

1kukFm1kvkFm2

m1 m2 eth49THORN


n22 kukGm





1kukFm1kvkFm2

m1 m2 eth51THORN


1kukFm1kvkFm2

m1 m2 eth52THORN


Slamp1



ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS


Sl Slamp1



frac14 Sl Slamp1

PlG



G


PlG





mn24

1 mnthorn24

2 kukFm1kvkFm2




where



1522 Sterbenz

ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



lmn12 kukFm

ethcmTHORN12kvkFl

c1lmn22 kukFm

kvkFl



12


Proof of the X12



kvkFl eth54THORN






kX1SlGkL1ethL2THORN

kX1SlGkX

12l1


in the X12



d1mn22 kvkFm

kukFl


ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS


cmd


Xcmd

md

12

mn22 kvkFm

kukFl







where

CI frac14Xdltcm


CII frac14Xdltcm


CIII frac14Xdm





eth56THORN


1524 Sterbenz

ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS





mY2x0x





ffiffidm




o1 o2 o3



bo1

llethdmTHORN12

sthornld so2

md

bo3

llethdmTHORN12

sthornld

eth57THORN





ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



eth58THORN



jx0 thorn xj


eth59THORN



So1



so1


ld

so2

md bo3

llethdmTHORN12

sld

eth60THORN





o1




p and


llethdmTHORN12


mdg the sum



1526 Sterbenz

ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS





so1


ld

so2

md so3

ld

eth61THORN


q and jo1 o2j

ffiffidm

q




12


p The result is


so1


ld

sm b

o3

lethldTHORN12slcm

eth62THORN


q




ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS





o


12

kvkFl

eth63THORN



Xo2 o3


kSo2


llethdmTHORN12


lXo


12 X

o

kBo


xTHORN

12

lXo


12

kvkFl




lXo2 o3


kSo2


llethdmTHORN12

SldvkL2ethL2THORN

lXo


12

kSldvkL2ethL2THORN

lmn22

d

m

n54

kukFmkvkFl

1528 Sterbenz

ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



12



d1Xo2 o3


kSo2


llethdmTHORN12


d1Xo


12

Xo

kBo


xTHORN

12

d12m

n22

d

m

n54

kukFmkvkFl




Xo1

kX1So1


12

ethldTHORN1X

o1 o2 o3

o1o3ethd=lTHORN12

o1o2ethd=mTHORN12

kSo1

ld

So2

mdu HSo3

ldv

k2L1ethL1THORN

0BBBBBB

1CCCCCCA

12

d1 supo



12

d

m

n54 d

l

n54

mn22 l

n22 kukFm

kvkFl





ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS

to compute that

Xo


12

ethldTHORN1Xo1 o3


kSo1

ld

Smu HBo3

lethldTHORN12Slcmv

k2L1ethL1THORN

0BB1CCA

12


kBo


12

md

12 d

l

n54

mn22 l

n22 kukFm

kvkFl



ACKNOWLEDGMENT


REFERENCES









1530 Sterbenz

ORDER REPRINTS



















ORDER REPRINTS



























Date post:	10-Jul-2020
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Global Regularity for General Non-Linear Wave Equations I ...bdriver/DRIVER/Miscellaneous... ·...

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