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TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 307, Number 1, May 1988 PSEUDO-DIFFERENTIAL OPERATORS WITH COEFFICIENTS IN SOBOLEV SPACES JURGEN MARSCHALL ABSTRACT. Pseudo-differential operators with coefficients in Sobolev spaces Hr'q, 1 < q < oo, and their adjoints are studied on Hardy-Sobolev spaces H8,p, 0 < p < oo. A symbolic calculus for these operators is developed, and the microlocal properties are studied. Finally, the invariance under coordinate transformations is proved. 0. Introduction. In [2], Michael Beals and Michael Reed developed a calculus for pseudo-differential operators with coefficients in Z/2-Sobolev spaces. They ap- plied it to microlocal regularity results for nonlinear equations and to the analysis of the propagation of singularities for quasi-linear partial differential equations. In view of these strong applications it is surprising how simple this calculus is. It de- pends only on some elementary estimates for the Fourier transform and for integral operators on L2. But this also explains that the method of Beals and Reed is not applicable to Lp,p ^ 2. In [3] Bony developed his theory of para-differential operators. In his paper the method of applying pseudo-differential operators with nonregular symbols to the propagation of singularities for nonlinear partial differential equations started. A little later Meyer [15] realized that the para-differential operators have something to do with the exotic Hormander class 5}",. It follows also from Meyer's results that the used pseudo-differential operators satisfy some Sobolev space estimates in the x-variable. In this paper we consider symbols a(x, £) which satisfy some uniform Hr'q- estimates in the ^-variable and the usual estimates in the ^-variable. Here the real numbers q and r are such that 1 < q < oo and r > n/q. We cannot treat the case 0 < q < 1, since we use, in an essential way, the decomposition into elementary symbols. Let us give an outline of the paper. In Chapter 1, we recall the definitions and basic properties of the function spaces we use, i.e. Besov and Sobolev spaces. In Chapter 2, we introduce the symbol classes 5™(r,q). The classes 5^(7,q) naturally appear in the study of nonlinear problems (see §4.3). However the case 6 > 0 is important for the development of the calculus in the later chapters. We study the behavior of the pseudo-differential operators and their adjoints on Hardy- Sobolev spaces Hs'p, 0 < p < oo. Also some estimates are given for operators Received by the editors October 6, 1986 and, in revised form, April 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 35S05, 47G05. Key words and phrases. Pseudo-differential operators, nonregular symbols, Sobolev spaces, microlocal analysis. The research was supported by the University of the Bundeswehr, Munich. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 335 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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Page 1: PSEUDO-DIFFERENTIAL OPERATORS WITH ...for pseudo-differential operators with coefficients in Z/2-Sobolev spaces. They ap-plied it to microlocal regularity results for nonlinear equations

TRANSACTIONS of theAMERICAN MATHEMATICAL SOCIETYVolume 307, Number 1, May 1988

PSEUDO-DIFFERENTIAL OPERATORSWITH COEFFICIENTS IN SOBOLEV SPACES

JURGEN MARSCHALL

ABSTRACT. Pseudo-differential operators with coefficients in Sobolev spaces

Hr'q, 1 < q < oo, and their adjoints are studied on Hardy-Sobolev spaces

H8,p, 0 < p < oo. A symbolic calculus for these operators is developed, and

the microlocal properties are studied. Finally, the invariance under coordinate

transformations is proved.

0. Introduction. In [2], Michael Beals and Michael Reed developed a calculus

for pseudo-differential operators with coefficients in Z/2-Sobolev spaces. They ap-

plied it to microlocal regularity results for nonlinear equations and to the analysis

of the propagation of singularities for quasi-linear partial differential equations. In

view of these strong applications it is surprising how simple this calculus is. It de-

pends only on some elementary estimates for the Fourier transform and for integral

operators on L2. But this also explains that the method of Beals and Reed is not

applicable to Lp,p ^ 2.

In [3] Bony developed his theory of para-differential operators. In his paper the

method of applying pseudo-differential operators with nonregular symbols to the

propagation of singularities for nonlinear partial differential equations started. A

little later Meyer [15] realized that the para-differential operators have something

to do with the exotic Hormander class 5}",. It follows also from Meyer's results

that the used pseudo-differential operators satisfy some Sobolev space estimates in

the x-variable.

In this paper we consider symbols a(x, £) which satisfy some uniform Hr'q-

estimates in the ^-variable and the usual estimates in the ^-variable. Here the

real numbers q and r are such that 1 < q < oo and r > n/q. We cannot treat the

case 0 < q < 1, since we use, in an essential way, the decomposition into elementary

symbols.

Let us give an outline of the paper. In Chapter 1, we recall the definitions and

basic properties of the function spaces we use, i.e. Besov and Sobolev spaces.

In Chapter 2, we introduce the symbol classes 5™(r,q). The classes 5^(7,q)

naturally appear in the study of nonlinear problems (see §4.3). However the case

6 > 0 is important for the development of the calculus in the later chapters. We

study the behavior of the pseudo-differential operators and their adjoints on Hardy-

Sobolev spaces Hs'p, 0 < p < oo. Also some estimates are given for operators

Received by the editors October 6, 1986 and, in revised form, April 6, 1987.

1980 Mathematics Subject Classification (1985 Revision). Primary 35S05, 47G05.

Key words and phrases. Pseudo-differential operators, nonregular symbols, Sobolev spaces,

microlocal analysis.

The research was supported by the University of the Bundeswehr, Munich.

©1988 American Mathematical Society

0002-9947/88 $1.00 + $.25 per page

335License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

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336 JURGEN MARSCHALL

whose symbols belong locally uniformly to Hr'q. We remark that related results

are presented in [4 and 22].

In the third chapter a symbolic calculus is developed. When composing two

pseudo-differential operators with nonregular symbols B and A the product B o A

is, in general, not an operator of the same class. Therefore, we decompose A =

Ay+A2 + A3. As in [15], this is a decomposition of the spectrum of a (•, £). Now for

B o Ay (resp. A*) we have a symbolic calculus whereas A2 and A3 are lower order

perturbations of Ay. This method has been already used for the class S™6(r, N) in

the author's thesis [12] and in [14]. Let us also mention [22].

Beals and Reed posed microlocal conditions on their symbols in order to apply

them to the study of propagation of singularities. In Chapter 4 we do the same,

i.e. we introduce the symbol classes S™(r, q) n S™lci(ri,<l', 7) and study their action

on the microlocalized Sobolev spaces Hs'p n H^fd)- m particular, we extend

Rauch's lemma (see [17]) to the full range 0 < p < 00. We also develop a calculus

for these operators and apply it to some results on microlocal ellipticity for nonlinear

differential operators.

Finally, in the fifth chapter, we prove that the classes S™(r,q) are invariant

under coordinate transformations. In fact, we prove a much more general result.

An appendix is devoted to some results about Hardy-Sobolev spaces which in

case 1 < p < 00 are due to Strichartz (see [19]).

Part of the material presented here is taken from the author's dissertation which

was written under the direction of Professor Karl Doppel. It is a pleasure to express

my thanks to him. I also would like to thank the referee for his valuable comments.

1. Preliminaries. Denote by 5 = 5(Rn) the Schwartz space of rapidly de-

creasing functions and by 5' = 5'(R") its dual, the space of tempered distributions.

The Fourier transform is defined on 5 by

(1) f(0:=Je-ix<f(x)dx

and extended to 5'(Rn) by duality. The inverse Fourier transform is

(2) />):=(2^/^/(£R.

Let the Bessel potential of order m E R be

(3) Jmf(x) := JL- J?*■*{! +\tfr'*f{Z)d£.

Let us define the Hardy-Sobolev spaces Ha'p for —00 < s < 00 and 0 < p < 00.

Choose <p E S such that f <pdx = 1 and let <pt(x) := t~n<p(x/t). Then for 0 < p <

00 the local Hardy space hp is the space of all tempered distributions such that

H/llftp := ||sup0<t<1 [ft * f\ ||_p < 00 (see [7]). Here <Pt * f denotes convolution.

One has hp ss U>, if 1 < p < 00 and h1 <-> L1.

Now the Hardy-Sobolev space Hs'p is the space of all f E S' such that ||/|| jj»,p :=

||J,s/IUp < 00. Next let p = 00. Denote by bmo the functions of bounded mean

oscillation

H/llbmo := sup — / |/(j,)|dy+ sup —- / /(_)-— / f(y)dy dx\q\>i\Q\Jq \q\<i\Q\Jq\ \Q\Jq

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 337

where the supremum is taken over all cubes Q; [Q[ denotes Lebesgue measure of

Q. The bmo-Sobolev space Hs°° is defined to be the space of all f E S' such that

||/||_M.~ == IkVllbmo < OO.The Schwartz space 5 is dense in Hs'p, if 0 < p < oo. The dual space is

(H3>p)' a H~s'p', if 1 < p < oo and 1/p + l/p[ = 1 (see [21, 2.11.2]). If Hs'°°

denotes the closure of 5 in Hs>°°, one has also (Hs'°°)' « H-3'1 (see [13]).

Note that the Hardy-Sobolev space is a quasi-Banach space. Instead of the

triangle inequality one has

(4) ll/ + ffllH..P<ll/|l«..P + llffllff..p» A = min{l,p}.

There is a Littlewood-Paley type representation of Hs,p. Denote by </>(Rn) the set

of all partitions {<pk} C 5(R") such that

(5) supp^C {£:!£!< 2}, supp^fc C {?: 2k~l < \f[ < 2h+1}

for k = 1,2,3,...,

(6) \da<pk(0\<Ca2-klal,oo

(7) £>*(£) = 1.k=0

(p(Rn) is not empty; see [21, 2.3.1.1].

Now for s E R and 0 < p < oo one has

/ oo \ 1/2

(8) ii/iih-p- £4*>*™a

Vfc=o / LP

(see [21, 2.5.8]). However, the corresponding statement for Hs'°° is false. In the

language of Triebel spaces (8) means that Hs'p is isomorphic to ^2-

We will also use Besov spaces Bp q. Let s E R and 0 < p, q < oo. Then B* g is

the space of all f E S' such that

/ oo \ V9

(9) II/I|b-„ == E 2fci«H^(^)/lllp < oo\fc=0 /

(modification for q = oo).

It holds that (0 < p < oo),

(1°) Bp,min{p,2} *-♦ HS'P *-» Bp,max{p,2}-

Estimation of the spectrum is a very useful tool when dealing with pseudo-differen-

tial operators on Hardy-Sobolev spaces. The following two lemmas are basic for

this purpose.

LEMMA 1.1. Let 0 < cy < c2 and fk E S' be such that

supp/o C {f: |e| < c2}, supp A C {£: c^*"1 < \f\ < c22k+1},

if k — 1,2,3,.... Then for each s E R and 0 < p < oo one has

oo / oo \ X/2

E/* ^c E4fcsiM2 • D

A:=0 h°.p \k=0 ) Lp

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338 JURGEN MARSCHALL

The lemma is an immediate consequence of the Nikol'skij representation (see

[21, 2.5.2]).

LEMMA 1.2. Let c > 0 and fk E S' be such that supp/fc C {£: |£| < c2k+1},

k = 0,1,2,.... Let 0 < p < oo and s > n ■ (max{l, 1/p} - 1). We then have

oo / oo \ V2

5> <c \Y:^\fk?)k=0 HS'P \k=0 ) Lp

PROOF. Let / = Y^T=o fk- There exists / = 1(c) E N such thatoo

<Pj(D)f= E <Pi(D)fk.k=]-l

Then with A = min{l,p} we obtain

/ 2\ Va A/CO OO

\\j\\h>*<c E4's E ^(^)a\j=0 k=j-l j

J _p

oo / oo \ 1/2 "

^CE E4J>^)W2fc=_i \j=0 /

Now using a vector valued Fourier multiplier theorem (see [21, 2.4.9]) we have

(E4^|^(_))/fc+,|2

^ ' LP

<cUp||^(2»"+fc.)II^J (E4JS|/fc+J|2)

^ ' LP

/oo V/2

<C2*(«-n/2) E4^|/„+J|2

^ ' £P

provided /c > n(max{l, 1/p} - 1/2). Here \\(p](2:,+k:-)[\H^,2 refers to the norm of

the function £ —» yjJ.(2-,'+fcf). Hence, choosing s + n/2 > /c, the lemma follows. □

Note that in case of Besov spaces there is an obvious counterpart for both lem-

mas.

We frequently use another characterization of Hardy-Sobolev spaces. Let tp be

a test function such that tp(x) = 1 for x E [0, l]n and let tpk(x) := tp(x — k) forkEZn.

THEOREM 1.3. Ifs>0 and 0 < p < oo, then

ii/iih- ~ ( E ii^/Hh-p )Wz" /

(modification for p = oo). □

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 339

PROPOSITION 1.4. LetO < q <p<oo and sER. Then we have

\\tpkf\\H^ < C||^fc/||H..P

with a constant C > 0 independent of k E Zn. □

In case 1 < p < oo both the theorem and the proposition have been known for a

long time; see [19]. In the general case they are proved in the author's thesis [12].

We give the proof in the appendix.

Let us also define the distributions which belong locally uniformly to Hs'p, i.e.

/e*Cififf

(11) \\f\\H'-> ■■= SUP \\rpkf\\H:P < OO.unif fcez"

By the proposition Hs'q «-» H^'pi{ for 0 < p < q < oo. Observe also that

MHS'P C H^'pi{ where MHS'P denotes the space of pointwise multipliers for i_5,p.

This explains some restrictions for pseudo-differential estimates given in the next

chapter.

2. Estimates for pseudo-differential operators.

2.1 The definition of symbols. Let m E R, 1 < q < oo, r > n/q, N E N and

0 < 6 < 1. Define S^(r, q; N) to consist of symbols a:R"xR"^C such that for

each |cv| < _V

(1) |^a(z,OI<C(l + |£|)m-N,

(2) \\dfa(;£)\[Hr.,<C(l + \S\r+Sr-W.

When N = oo, write S^(r, q) instead of S^(r, q; oo). To each symbol a E S^(r, q; N)

associate a pseudo-differential operator

(3) Op(a)f(x) := —^ J e^a(x, 0/(0^

for f E S. Let us then simply write Op(a) E S™(r,q; N).

In the following we shall always assume that (1 — 6)r > n/q. This is the counter-

part for the condition 6 < 1, when q = oo. Observe that by the Sobolev embedding

theorem

(4) Sr(r,q;N)cSZ(ry,qy;N)

ior 1 < q < qy < oo, ry = r — n(l/q — 1/qy) and £i = 6(r/ry). Note that

(1 - <5i)ri - n/gi = (1 - S)r - n/q.

Call a E Sj-(r, q; N) an elementary symbol, if a = Yl'kLo Mk(x)ipk(£) is such that

(5) suppt/.oC{£:|e|<4}, supp^C {e:2fc-2<|,e|<2fc+2}

for k = 1,2,3,..., \datpk\ < C2~k^ for all multi-indices a such that |q| < TV, and

if |Mfc(x)| < C, \\Mk\\Hr.<, < C2Ur. The point is that any symbol a E S$(r,q) can

be decomposed into elementary symbols.

PROPOSITION 2.1. LetaE S$(r,q). Given 0 < X < 1 and N E N there exist

a sequence {ck} E lx(Zn) and elementary symbols ak E S$(r,q;N) such that

(i) a = E*_z" °kak,

(") \Wk\\s°(r,a;N) < C||a||S0(r,<,) • □

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340 JURGEN MARSCHALL

Such decompositions have been introduced in [5, Chapter 2.9]. The proof of

the proposition is essentially the same as that one given by R. R. Coifman and

Y. Meyer. It depends only on the fact that the Sobolev spaces Hr,q are Banach

spaces, if 1 < q < oo. For this reason we are unable to treat the case 0 < q < 1.

2.2 The main estimate. For a real number x define as usual x+ := max{0, x}.

Then the main result in this section can be stated as follows.

THEOREM 2.2. Let AE Sf(r,q) be such that 1 < q < oo, (1 - 6)r > n/q andm E R. Let 0 < p < oo. Then for each real number s such that

n(l/p + 1/q - 1)+ - (1 - 6)r < s < r - n(l/q - l/p)+

the operator A: Hs+m'p —» Hs'p is bounded. If, in addition, (1 — 6)r > n/q, then

the same is true for s = r — n(l/q — l/p)+. □

In the case p = g = 2, <5 = 0 and some values of s E R the above theorem is

found in [2]. The case q = oo is settled in [12]. The main ideas of the proof of

Theorem 2.2 are already presented there. Let us mention also [4 and 22]. Note

that for elementary symbols the theorem holds even when 0 < q < 1.

The proof of the theorem will be given in several steps.

2.3 A first estimate. We consider first the case s < r — n(l/q — l/p)+ and p < oo

of the theorem. In this case the theorem holds for more general symbols. Suppose

that for each multi-index a

(6) \d%a(x,Z)\<Ca(l + \Z\)m-H,

(7) ll^a(-,e)||B;i00<Ca(l + |e|r+6r-lQl

where 1 < q < oo, r > n/q. Hence, we allow the case r = n/q.

THEOREM 2.3. Suppose a satisfies (6) and (7) above, and suppose further

(1 - S)r > n/q. Then for 0 < p < oo and

n(l/p + 1/q - 1)+ - (1 - 6)r < s < r - n(l/q - l/p)+

the operator Op(a): Hs+m'p — Hs'p is bounded.

PROOF, (i) Let m = 0 and suppose that a = Efclo Mk(x)ipk(tl) is an elementary

symbol, that is, suppose that ||Mfc||L°° < C and HM/tHBj^ < C2kSr. In fact, the

counterpart of Proposition 2.1 is valid. Further, in view of (1.4), we may choose

A = min{l,p}. Hence it is no restriction to assume that o is an elementary symbol.

We may also suppose that

[datpk\ < CQ2-fc|Q| for all |a| < N, N > n(max{l, 1/p} - 1/2).

We then have for s E R and 0 < p < oo

/ oo \ V2

(8) [J2^s\MD)f[2) <C||/||h..pWo / LP

(see [21, 2.4.9]).

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 341

(ii) Decompose

oo fc—4 oo fc+3 oo oo

a = E E M^k + E E M^k + E E Mfe^fefc=0>=0 fc=0j=fc-3 fc=0j=fc+4

= ay + a2 + a3

where Mkj := <pj(D)Mk for some {v^} £ <^(Rn).

Define fk := tpk(D)f. The spectrum of YljZo ^kjfk is contained in the annulus

[n\ ~ 2k. Hence, by Lemma 1.1

/ 2\ V2/ oo fc-4

\\Op(ay)f\\H,,<C E4*8 EM^*I fc=4 j=0

(9) / oo \ 1/2

<c E4fcsi^2

Vfc=4 / L„

< C\\f\\H..pfor each real number s.

(iii) We estimate Op(a2)/. Suppose first q = oo. Since then ||A_kj||_oo <

C2-jr+kSr^ we obtain by Lemma 1.2

/oo y/2

||Op(a2)/||H.+(,-s„.p<C (E4/CSIM2)\fc=o / LP

<C\\f\\„s,,

if s > n(max{l, 1/p} - 1) - (1 - 6)r.

Now let 1 < q < oo. We use the embedding

fill as+n(l/p,-l/p) <_^ rrs.p r>s-n(l/p-l/p2)V11/ Pi,P P2,P

for 0 < pi < p < p2 < oo (see [6, 9 or 13]).

Let 1/pi = 1/q + l/p2, Pi < p < P2 and observe that ||A_fcj||_<, < C2-jr+kSr.

Then (11) and Lemma 1.2 for Besov spaces yield

l|Op(a2)/||H» + (1-4)r-"/'».P ^ C||Op(a2)/||n.-r(l-«)r-F.(l/p-l/p2)

(12)< C||/||R.-r.(l/p-l/p2) < C||/||ff».P,

aP2,P

if s > n(l/p + 1/q - 1)+ - (1 - 6)r. To see this choose 1/pi = 1/p +l/q-e, e > 0

arbitrary small.

(iv) For q = oo we have by Lemma 1.1

( a 2\ 1/2oo 3-4 \

E4js EM^/*3=4 k=0 J

(13) /oc /y-4 xV/2

<C7 E4^-)(E2^l/fclJ^ ' LP

^ C'||/||tf,-(l-«)r,p

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342 JURGEN MARSCHALL

In case 1 < q < oo suppose first p < q. Then, choosing 1/pi = 1/p + e, we obtain

similar to (12)

(14) ||Op(a3)/||jf..P <C\\f\\H,-^-s)r+n/q.P

when s < r.

In case p > q observe that

\\Mk\\B;7«v,-v» < C||Mfc||Bri00 < C2kSr

which allows us to remove the restriction p < q. Hence, (14) holds, if s < r —

n(l/q-l/p) + . D

We remark that in case q < oo Theorem 2.3 is true for arbitrary Triebel spaces

Fpp.,0 < p* < oo (compare [13, Theorem 3]). In case q = oo and 6 < 1 see [12,

Theorem 3.1]).

2.4 The adjoint estimate. Let the adjoint operator A* be defined by

(15) j Afgdx= f fA^gdx, f,gES.

Then the counterpart of Theorem 2.3 is

THEOREM 2.4. Suppose a satisfies (6) and (7) above, and suppose further

(1 — b~)r > n/q. Then for 0 < p < oo and

n(l/p + 1/q - 1)+ - r < s < (1 - 6)r - n(l/q - l/p) +

the operator Op(a)*: Hs'p -* Hs-m<p is bounded.

PROOF. Let m = 0. The case 1 < p < oo follows from Theorem 2.3 by duality.

Hence, suppose that 0 < p < 1.

(i) The adjoint of Op(ai) is given by

oo /fc-4 \

Op(a1)*ff = EV'fc(^) E*& •fc=4 \j=0 J

Here we have gk := ip'k(D)gk for a suitably chosen smooth function tp'k supported

in the annulus |n| ~ 2k. Hence, by the vector valued Fourier multiplier Theorem

2.4.9 [21] we obtain

/ 2\ i/a/ oo fc-4

||Op(a1)*ff||J...p < C E4"S E^^^lb^ lfc=4 j=0

^ 'LP

<C||„||H..p

for each s E R.

(iii) The adjoint of Op(a-2) is given by

3 oo / fc+8 \

Op(a2)*S= E E^(P) Mfc^E^ •i=_3fc=o y j=o J

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 343

Again, by the vector valued Fourier multiplier theorem we get, if q = oo and

s< (l-6)r,

{ n i/2/ oo fc+8

■ [\Op(a2yg\\Hs,P <C E ±k(s-(1-8)r) E 9i

^'> lfc=0 j=0^ ' LP

^ C||<7||_?»-(i-<o--.p.

In case 1 < q < oo combine the ideas leading to (14) and (17) to obtain

(18) ||Op(a_)*_r||__-,p <C||^||ffs-(i-«)r+n/,,p,

if s < (1 - 6)r (recall that 0 < p < 1).

(iii) Finally the adjoint of Op(a3) is

oo oo

Op(a3)*0 = E E MD)(Mk~393).k=0j=k+4

Here the vector valued Fourier multiplier theorem yields in case q = oo,

oo / oo _ \ i/2 p

l|Op(a3)*.||^.+(l_S),p < CE E*ka\MD)(Mk-k-+~9k+3)?3=1 \k=0 J Lp

foe \ 1/2 "

<cJ223iK-r-s)p E4("+J>l9fc+,fj=4 \fc=0 / Lp

for some n > n(l/p — 1). Hence, if s > k — r, we get

(19) ||Op(o3)*ff||tf.+(i-«)r.p <C\\g\\H:p.

Finally, in case 1 < q < oo, combining the ideas leading to (14) and (19) yields

(20) ||Op(a3)*g||ff.+(i_«)r-n/,,p <C\\g\\H>.p,

ii s > n(l/p + 1/q- 1) -r. O

Let us point out that the remark following Theorem 2.3 applies word for word.

2.5 Proof of the main theorem. We are now in the position to prove Theorem

2.2.

PROOF OF THEOREM 2.2. (i) Consider first the case

n(l/p+l/<7-l)+ -(l-<5)r < s < r - n(l/q - l/p)+.

When 0 < p < oo this case follows from Theorem 2.3, and when p = oo from

Theorem 2.4 by duality.

(ii) Let now s = r — n(l/q — l/p)+. In view of (4) we may suppose that p < q.

Assume first 1 < q < oo. Then defining A := min{l,p} and 1/p = 1/q + 1/py we

obtain

Aoo oo oo

||Op(a3)/||k,P < E E M>vfk <Cj2(\\Mk\\Hr,q\\fk\[LPl)xfc=0 3 = k+4 Hrp fc=0

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344 JURGEN MARSCHALL

and consequently

(21) ||Op(a3)/||^.P < C\\f\\B*r < C||/||fl*r+„/, .p,,rcim{l,p} p,min(l,p}

In case 0 < p < 1 we get by (12) even

(22) ||Op(a3)/||//r,P<C||/||ffSr+„/„p.

(iii) Consider finally the case s = r and q = oo. When 0 < p < oo observe that

by Proposition 1.4 __r,°° C #„„;'* for each qy < oo. This case then follows from

Theorem 2.6 below.

When p = q = oo observe first that for r > 0

(23) \\f-g\\Hr.<» <C(||/||Lc0||S||H„c0 + ||/||i/r,oo||ff||_«).

Hence, Hr<°° is a multiplication algebra. In fact, we have __^0 5_vio fj9k = Ag

where A is such that

\d$d$aA(x,(i)\<CaML~(l + \li\)W-W.

Hence A E 5?(oo, oo), and we obtain

oo fc+3

EE-^* <C\[f\\L°°\\g\\Hr.°°.fc=oi=o Hrt00

Similarly one shows

oo oo

E E fiSk <C\\f\\Hr.-\\g\\L-.k=0j=k+4 Hroc

Now (23) implies

oo oo

||Op(a3)/||i/r.co <J2 Yl M"if"fc=0 i=fc+4 Wroo

oo / oo oo \

^CE E M* ii/*iitf'.~+ E M*i iiMU- •fc=0 \ j = fc + 4 LOO j=k+4 Hri00 /

Then || E°lfc+4Mfc>||_~ < C2-*'||M„||BSooo < C2-*C1-«>' yields

(24) l|Op(a3)/||_fr.co<C||/||fl^i.

This completes the proof of the theorem. D

Before proceeding to Theorem 2.6 let us single out an inequality which will be

useful in the next chapter.

For a E S™ (r, q) and N E N sufficiently large define

llalUcc := sup sup (1 + \Z\)~m +|a||^a(_, 01(*,€)M<JV

and

||a||„r., :=sup sup (1 + |_irm-«r+|a|||3?a(-,0llH'-«-<; |a|<Af

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 345

We then have, if 1 < q < oo,

(25) ||Op(a)/||ffr., < CdlalU-ll/H^+m., + ||a||„r„||/||B.r+m).oo,l

This inequality follows from (9), (12) and (21) in case 1 < q < 00 and from (16),

(17) and (24) in case q = 00.

Let us give another inequality of this type. If 0 < p < q < 00, n(l/p+l/g-l)+ <

2s, we have in case 1 < p < 00

II. • /||i/..p < C(||_||_~||/||„..p + llffll/f .«II/IIB-/.)

and in case 0 < p < 1

\\9 ■ /||_r..P < C(||ff||Lc.||/||i...p + ||ff||j_.„.||/||ff-/,.p).

Note that the case 0 < q < 1 can be proved just as the case 1 < q < 00. Then

Theorem 1.3 yields

(26) [[g ■ f\\H.,, < C(\\g\\L~ ||/||*..p + \\g\\H'u«t ll/ll_r-/.+.*)

for each _ > 0 when 1 < p < 00 and

(27) ||9 • f\\B.„ < C(||.||_=c||/||tf..p + ||ff||H..,f ||/||„-/,.p)

when 0 < p < 1.

Now recall that Hs'p C L°° iff s > n/p and 0<p<lors> n/p and 1 < p < 00

(see [9]). This gives us part (a) of

THEOREM 2.5. (a) Let 0 < p < 00 and s > n/p, ifO<p<lors> n/p, if1 < p < 00. Then we have MHS'P = H^pi{.

(b) If r > 0, Hr'°° is a multiplication algebra. □

In fact, (a) follows from (26), (27) and the method of Strichartz [19]. (b) is

merely a restatement of (23). Observe that by Theorem 1.3, H^{ = Hr'°°, r>0.

The inequalities (26) and (27) allow us to treat operators with symbols in H^i{.

Suppose a satisfies for each a

(28) [data(x,f)\<Ca(l + \f\r-W,

(29) ||3f _(-, OlUc, < Ca(l + leir+^-M

THEOREM 2.6. Suppose that a satisfies (28) and (29) and that (1 -6)r > n/q.

Then for 1 < q < 00,0 < p < 00 and

n(max{l, 1/p} - 1) - (1 - 6)r + n/q < s <r - n(l/q - l/p)+

the operator Op(a): Ha+m'p -> Ha'p is bounded.

PROOF, (i) We may suppose m = 0. The decomposition into elementary sym-

bols remains valid. Hence, suppose a = YlkLo ^fc(I)V'fc(0 with ||Mfc||_«> < C and

\[Mk\[Hr, <C2k6r.unit

The estimation of Op(ai) remains the same. Moreover, since (5^^ )unif =

Btt,olq, we obtain similarly to (10)

(30) ||Op(oa)/||j..+(i-«)r-/M < C||/||«..p.

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346 JURGEN MARSCHALL

(ii) For the estimation of Op(o3) we may suppose that p < q. Now suppose

n(max{l, 1/p} - 1) - (1 - 6)r + n/q < s < r and n(l/p + 1/q - 1)+ < 2s. This is

possible at least for s = r. Then from (26) and (27) we get

oo / oo \

E Mkofk <C2knlq 2*<-(1-fi>r> + 2ta E Mki IIMIlp-3 = k+4 _ I j=k+4 H,,q J

V unify

Now use the following lemma which is an easy exercise in smooth pseudo-

differential operators.

LEMMA 2.7. IfbES^o, then Op(b): H^'p - H^l{• □

Defining 6(0 := £°!lfc+4<p.,-(0 we have

\dabiO\ < C2k^s'r^(l + |£|)s-r-H

and hence, by the lemma

oo

E Mkj <C2^-(l-S)r)^

j = fc+4 „Slqunif

This certainly yields for each e > 0

(31) ||Op(a3)/||//3,P < C||/||H.-(l-«)r+n/,-r«,p.

Finally we have to remove the restriction 2s > n(l/p + 1/q — 1)+. Define, for

s < 0, b^(n) :— (1 + |f + n\2)s^2tp^(n) where tp^ is a smooth function such that

tp^(n) = 0, if \n\ < 2 ■ |£| and tp^n) = 1, if |n| > 3 ■ |cj|. Then obviously

I W>)l < ca(i + if/D-w < ca(i + \$\)s(i + M)-H

Now the operator Js o A3 has the symbol a(x, 0 — Op(b^)a3(-, 0- But then the

lemma yields

hi;0IIh-« < c(i + leiriM-,oil//-, < c(i + \t\y+6r.unit unit

Consequently, if s < 0, the operator Js o Op(a3) o J~s has a symbol which satisfies

(28) and (29). Hence, (31) is true for arbitrary s <r. O

3. The calculus for pseudo-differential operators.

3.1 The spectral decomposition of symbols. M. Beals and M. Reed developed in

[2] a calculus for 5o"(r, 2), r > n/2. Their main tools are some simple estimates

for the Fourier transform. A different method was used in [12 and 14] to develop

a calculus for S™g(r, N). Decompose a symbol into a smoother one and a lower

order perturbation. For the smoother symbol a symbolic calculus can be developed

by using pseudo-differential estimates. It is easy to adapt these arguments to the

present situation.

The decomposition of the symbol which we have in mind is a decomposition of

the spectrum of the function x —> a(x, 0- This has been used earlier in [5 and 15].

A somewhat different decomposition is used in [11].

Let a E S?(r,q). We define a E S^(r,q) iff

(1) suppa(.,Oc{r/:|r/|<^(l + |e|2)1/2},

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 347

aES^(r,q)iS

(2) suppa(., 0 C {rf. jfofl + \t\2)1/2 < [v\ < 100(1 + |e|2)1/2},

a€S6m3(r,9)iff

(3) suppa(.,Oc{r,:50(l + |e|2)1/2<|?7|}.

Let us also define Op(a) E S$(r, q) iff a E Sfi(r, q).

This is essentially the same decomposition as in the proofs of the preceding

chapter. The numbers ^, ^ etc. are chosen only for convenience. Any other

numbers c •< l,c/2 etc. will do the job as well. Any symbol a E S™(r,q) can

be decomposed into a = ay + a2 + a3 such that ai E 5^(r,g). For elementary

symbols this is done in the proof of Theorem 2.3, and the general case is similar. A

fundamental property of the symbol classes Spi (r, q) is that the elements of 5^\ (r, q)

and S™2 (r, q) are smoother whereas the elements of S™2 (r, q) and 5™3 (r, q) are of

lower order.

LEMMA 3.1. Let (l-8)r >n/q.

(a) For 0 < 8 < 8' < 1 we have

S^(r,q)cSP([(l-6)/(l-6')]r,q)

(if 6' = 1, let [(1 - 8)1(1 - 8')}r = oo).

(b)S^(r,q)GSr{1-S)T+n/Qioo,9)-

(c) IfO < r < r is such that (1 — 6)(r — r) > n/q, then

S^2(r,q) + S^3(r,q)cSr{1~6)T(r-r,q).

(d) If 8' > 8 is such that (1 — 8')r > n/q, then

S^2(r,q) + S^(r,q)cS^-{S'-6)r(r,q).

PROOF. Let a E 5£\(r,<j) and r' = [(1-8)/(l-8')]r. Then using an inequality

of Plancherel-Polya-Nikol'skij type (compare [21, 1.3.2.1]) yields

hi, o\\Hr', < c(i + ki)r'-rH-, oik- < c(i + ieir+*v.

Hence, we obtain (a). Next, let a E S™2(r,q) + S™3(r,q) and define ^(-,0 :=

(Pj(D)a(-, 0, {<Pj} E (p(Rn). We then have

K(x,OI < C2-^-"/*)||a(.,0l|s;rS/, < C2"^-"/*)(l + mr+sr,

and hence summation over all natural numbers j such that 23 > 1 + |£| yields

|a(„,oi <c(i + |c;|)m-(i-*>r+n/«.

Now (b), (c) and (d) follow easily. □

In the preceding chapter we decomposed an elementary symbol into a = ay +

a2+a3. This corresponds to the decomposition of a such that a^ E S^r, q). Hence,

any estimate for Op(a^) obtained in the preceding chapter yields an estimate for

the symbol class S™(r,q).

Observe also the following. If a € 5™1 (r, q) and 6 E 5™2 (r, q), then we have

(4) a-bES^+m*(r,q), 8 = max{8y,82}.

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348 JURGEN MARSCHALL

This is an immediate consequence of the inequality

11/ • 9\\h^i < C'II/HlooIIsII//- + II/II//'.''IMIl°°.

3.2 Composition of operators. When constructing a symbolic calculus for pseudo-

differential operators, for example the S™6-calculus in [8], the remainder term

(5) C(X, 0 = OBoA ~ E ~sdtbD>*—' a\ *

has usually the order my + m2 — (1 — 8)(l + 1). This is no longer true when one

is working with symbols in Hr'q, 1 < q < oo. In fact, the decrease of the order

depends on the i_ r*,00-regularity of the symbol. By (2.4) we have r* = r — n/q and

8* = 8r/r* = 8(r/(r-n/q)). We will see that c has order my+m2 — (l — 8*)(l + l).

Observe also the following. If a € S™(r,q) and \a\ < r — n/q, then we have

D%a E S£+4*|a|(r - \a\,q), 8a = (Sr - 8*\a\)/(r - \a\). This is in contrast to the

case q = oo. Note that 8a = 8* = 8, ii q = oo or 8 — 0. In case r = oo one has

8* = 8 + e, e > 0 arbitrary small. We don't know if we can choose in this case

8* = 8.Now we give a more precise meaning to (5).

THEOREM 3.2. Let B E S£2(r2,q) and A E S^(ry +r,q) be such that 0 <

fi < r2> (1 _ o~i)fi > n/g and 0 < r < I + 1. Then for

C(X, 0 = °BoA - E -&hD>*—' a\ s

\a\<l

we have

c E Sri+m2~{1~6:)T(f2,g), 8 = max{l - (1 - __)n/r,fc}.

PROOF, (a) By the Taylor formula we have

c(*.0 = ^ E -J\l~t)1 fe^n"d<Zb(x,f + tn)a(r,,f)dridt.

Let tp E C°° be such that tp(n) = 1 for \n\ < l/20,tp(n) = 0 for \n\ > 1/10 and

define tp^(n) := tp(n/(l + |c;|2)1/2). Observe that a(n, 0 = tp^(r])a(n, 0- View f as

a parameter and define

da(x,v)~Vadaib(x,ti + tn)iP(:(n).

We then have |f + tn[ ~ |£| and therefore

(6) \dpnda(x,v)\ < C(l + K|)m2-(i+1)(l + \V\)1+1~W,

(7) ||c^dQ(.,n)||*r2„ < C(l + |£|r*+^-«+D(i + [n\)l+1-W.

Hence, we may view da as an element of 50+ (r2,q).

(b) Define c_(-, 0 := Op(d_)a(-, 0- From (2.25) we obtain

||c«(-,OI|jir'...-<C((l + |€|)m»-<,+1)||o(-,e)l|j.'.+«+M

+ (i+ieir2+62r2-(m)iK-,e)iiB'+\)-oo, 1

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 349

We have

H; oiiw+M < c(i +1 z\y*+l+1-r*-T\\a(; o\\H^+r,< (j{y^ i |t|'\»ni+ra-r.|+l-(l-6i)(ri+r)

and

<c(i + ^|)mi+'+1_(1-*i)r

with a simple modification, if r = 0. Hence, we get

(8) \K(; 0||//^., < C(l + mri+m2+S2r2-(l-S-1)r_

(c) Let {(fj} E </>(R") and define o,-(-,0 := v?j(-D)a(-,0- We then have

|CQ(X,0I< E \Op(da)aj(x,f)[.

Now using the Plancherel-Parseval theorem and the Bernstein inequality we get

\Op(da)aj(x, 01 < cyv+v J \d%b(x, e + tn)Wi)*i(i> 0| */

<C72»'(,+1)||a?6(»,€-+*-)^(-)||A-/.||oi(.,OI|ji-

(compare [15]). Recall that \\f\\Bn/2 < C\\f(T-)\\Bn/2, T > 0. Hence, the choice

T=^(l + |e|2)1/2 yields

|Op(dQ)a,(_,OI<C2^+D(l + |N;|r2-('+1)|K(.,OI|L»,

and we obtain

|ca(*,OI<C(l + |Cir«-('+1)||a(.,OII_H+i(9) °°-1

< c(i + |^|)TOi+m2-(i-*r)T.

Now the theorem follows. D

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350 JURGEN MARSCHALL

An immediate consequence is

COROLLARY 3.3. If BE S£2(r,q) andAE S™\(r,q) such that (l-8i)r > n/q,then

BoAES^l+m2(r,q), 8 = max{6y,82}. □

As a typical application of the theorem let us prove a result about commutators.

For simplicity let us consider the case 8 = 0.

COROLLARY 3.4. Let B E 5o"2(r,<j), A E S™l(r,q) andletr + n/q < r, r < 1.

Then for each real number s such that

(1 1 \ + (1 1\+nl —I-1) — r + r + max{0, — m2} < s < r — n I-1 — max{0, m2}

\P 9 J \9 Pj

the commutator Bo A- Op(ba): Hs+mi+m2-T<p -> Hsp is bounded.

PROOF. Decompose a = ay + a2 + a3 such that ai E S™1 (r,q). Then by the

theorem oboAx - Op(6ai) E S™1+m2~T(r,q) for 8 = r/r (choose ry = r — r).

Moreover, by Lemma 3.1 (d) and (4) we get 6(a2 + a3) E 5™1+m2_T(r,q). Hence,

noting that (1 — 8)r = r — t we obtain the boundedness of B o Ay — Op(ba).

Finally, the boundedness of B o (A2 + A3) follows from the results of Chapter

2. □Observe that the proof of the corollary shows that it is useful to have estimates

for S™(r,q), 8 > 0, even when one is dealing with 5o"(r, q).

3.3 Adjoint operators. Let a E 5™, (r, q) and define

(io) c^o-^.-^iaf^.i_i<i

Then by Taylor's formula

c*(*>0 = ̂ r E A i\i-t)1 f f e^-y^(dlD-a)(y,P + tn)dydr,dt.(27r) |_m+i a Jo J J

Note that by the condition on the spectrum of _(■, 0 the integration is performed

over all n such that |«7| < jo\t]\. For those n we have obviously

\dPd](d%D2a)(y, f; + tn)\ < C(l + |f |)m"(i-«*)r+H-l/J|

provided r is such that 0 < r < I + 1 and n/q + r < r. Hence, partial integration

with respect to the operator

{(1 + (f)2\x - yl2)"^ + (O-'M2)-1*.! - <a2A„)(l - (0"2A,)}"

where (0 := (1 + |C|2)1/2 yields

(11) \c*(x,0\<C(l + \f\)m-^-s')T.

Observe also that

c*(V, 0 = (/ + 1) E ~, I C1 - t)'d^a(V, £ + tn)dt

implies

(12) suppc*(.,Oc{n:|n|<^(l + |el)}.

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 351

Hence, we obtain

THEOREM 3.5. Let A E 56m,(r, <?) and r be such that (1 - 8)r > n/q, 0 < r <

/ + 1 and n/q + r < r. Let c* be defined by (10). Then for each real number s and

every 0 < p < oo the operator Op(c*): /p+m-(i-6">.p __ jjs,v is hounded. D

This theorem is in many situations good enough to handle adjoints. For example,

one may prove

COROLLARY 3.6. Let A E S^(r,q) and 0 < r < 1 be such that n/q + r < r.

Then for 0 < p < oo and

(1 1 \+ (1 1\+n[ - -\-1 - r + r + maxiO, -m} < s < r - n- - max{0, rn}

\P 9 J \9 P)

the commutator A* - Op(a): Hs+m~T'p -» Hs'p is bounded. D

Let us remark that in case m > 0 the corollary holds for s = r—n(l/q-l/p)+ -m,

whereas in case m < 0 and 1 < p < oo it holds for s = n(l/p+l/q-l)+ -r + r-m.

This follows from the estimates for Op(<i2 + ^3) and A2 + A3 given in Chapter 2.

4. The microlocal calculus.

4.1 Microlocal symbols. Let (xo>£o) E R" x 5n_1. We say that a tempered

distribution / belongs microlocally at (xoi£o) to Hr'p, f E -^^((zoi £0)), iff

tp(D)(ipf) E Hr'p for a test function <p such that <p(x0) ^ 0 and a symbol tp E 5°0

such that V(0 = 1 for c; 6 c(£o,e) := {n: [n\ > 1 and (n,cjo) > (1 - e) • \v\} for

some 0 < _ < 1.

Let 7 C Rn x 5""1 be closed. We say / € #^,(7) iff / G Hrmpcl((x0, &)) for ev-

ery (x0, tio) E 7. For abbreviation let us write \[f[\H^cl((xo,M) := IIV'(£))('P/)||//^p.

When the point (zrj)£o) is understood we shall also write ||/||Br.p instead of

11/11//-, f •Now let a E S^(r,q) be a symbol. We say o E Sj?(r,q) n S™_.(ri, .n) iff for

every (xo, tio) E 7 and every multi-index a

(i) iia?a(-,oiij.'»-<c(i+ieiriHa|' mcl

where the constant C > 0 may depend on (xo, <;o) and a, but not on £.

In case 8=0 and q = 2 such symbols are introduced in [2] for the study of

propagation of singularities for nonlinear problems. There the regularity assump-

tions in the Ovariarjle are very weak. However, this is special to the case q = 2.

Of course, even in case q ^ 2 our smoothness assumptions in the Ovariable can be

weakened (though not as weak as in the case q = 2).

When acting on Hm+S>p n H™+ruV(1) the operators in S^(r, q) n S^,{ru q; 7)

have very good continuity properties.

THEOREM 4.1. Let a E Sp(r,q) n S™lcl(ry,q;~i) be such that 1 < q < 00,

(1 — S)r > n/q, my < m + r\ — (1 — S)r. Let 0 < p < q and ry > n(max{l, 1/p} — 1).

Then f E //w+n-(i-«)r+«/«J» n H^""^) implies

Op(a)f E H°<p n IQ*(7), s := min{r, n - (1 - 8)r + n/q}.

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352 JURGEN MARSCHALL

PROOF, (i) Let (xo,tlo) E 7. Let tp and tp be as above such that tp(D)(tpf) EHm+r,,p and

(2) \\d?tP(D)(v ■ _(•, 0)||//m. < C(l + |e|)mi-|a|-

Suppose that tp(£) = 1 for £ € c(£o,e). Suppose also <p(x) = 1 in a neighborhood

of Xo-

In the following replace a(-, 0 be tp ■ a(-,0- Hence, a(-,0 is supported in a

neighborhood of xq- Then decompose a = ay + a2 + a3 such that a; E 5™j(r, q).

We may suppose in addition that

(3) suppa1(-,0c{r>:M<(e/3)|£|}, 8uppa3(-,0 C {r/: |r/| > (3/_)|£|}.

(ii) Consider first Op(ay)f. The operator g —► Op(ay)(tpg) has the symbol

E -,Dap(x)d?ay(x,0 + fN(x,l;);^—' a\

\a\<N

hence,

p • Op(0l)/ = Op(ay)(rf) - rN(x,D)f modHr^p((xo, £o))-

Now consider

Op(ffll) (vfnr,) = J ay (n - f, 0 (p/H .)<*£■

Then by (3) (£, &> < (1 - e) ■ |£| implies

fa,_o)</H*|, /? := (1 - §e)/(l - f).

Hence, if tp E 5°0 is supported in the cone {n: (n, £0) > /? • |n|}, we obtain

tP(D) o Op(ai)(^/) = tP(D) o Op(ay)(tP(D)(tpf))

and consequently

(4) \MD) O 0P(a1)(^/)||Br1.p < CWfWjjm+r^.

Further, r^ (_, 0 is a sum of symbols of the form

Jeix«d%ai(x, £ + tn)(Davr(n)dn.

Now recalling Peetre's inequality (1 + |£ + tn\)x < C(l + |£|)A(1 + |n|)|A| and

observing that (Datp)~is rapidly decreasing we obtain

M*,0i<c(i+i_ir-(JV+1).

The same argument shows r^ E 5™ + (00,00). This implies for N + 1 >

(l-S)r- (n/q),rN(x,D)f EHri'p. Hence, we get

(5) ||Op(a,)/||wn p < C(\[f[\Hm + r,.p + ||/||7fm + r1-(l-«>r-W«,p)-mcl met

(iii) Since ry > rc(max{l, 1/p} — 1), we get by Lemma 1.2

(6) ||Op(a_)/||//M.P < C\\f\\Hm + r1-t.l-S)r + n/q.p.

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 353

(iv) Similarly to the discussion in step (ii) we get tp(D) o Op(a3)/ = tp(D) o

Op(tp(D)a3(-, 0)/- Since obviously

||V(Z))a3(.,0ll//n„<C||a(.,0ll//nfmcl

(recall that a is supported in a neighborhood of xo), this yields by (2.21)

(7) ||0P(a3)/||if'l.P<C||/||B«I+„/, <q|/||ffm+r1-<l-«>r+„/„P,mcl ■Dp,min{l,p>

since my < m + rx — (1 — 8)r.

(v) Since by step (iii) Op(a2)/ E HruP, we get Op(a)/ E Hs'p. U

Let us state two important corollaries of the theorem.

COROLLARY 4.2. Let a E Sp0. Then f E Hm+S>p n H^^d) implies

Op(a)fEH^pnHrmpcl(l)- □

Note that there are no restrictions on s and r. The reason is that Op(a2 +a3) EQ — OO°i,o •

COROLLARY 4.3. Let 0 < p < oo and n/p < s < r < 2s - n/p. Then

Hsp n /f^j('y) is a multiplication algebra. D

The case 1 < p < oo follows directly from the theorem. The proof of the case

0 < p < 1 is the same, because no decomposition into elementary symbols is

involved (see the remark at the end of §2.2). The case p = 2 of the corollary is

known as Rauch's lemma; see [17 and 1].

4.2 The calculus. When developing a calculus for S™(r,q) D 5™^ (ry, q; 7) we

proceed as in Chapter 3. Decompose a symbol a E S™(r,q) fl S™^ (ry, q; 7) into

a = ay + a2 + a3 such that a; E S^(r,q). Then by Corollary 4.2 a* E S^(r,q) f)

5^,^1,(7; 7). Observe also the following counterpart for inequality (2.25) which is

an easy consequence of the proof of Theorem 4.1:

||Op(a)/||i/M.;<C(|H|L-||/||Hm+r1.,/o\ met ilmcl

+ ||a||//''.')||/||Bm + r1-(l-«)r+n/,,, + ||a||Hr, , || /|| gm , )mcl 00,1

where, of course,

\\a\\Hry, :=sup sup (l + |e|)-mi + |a|||^a(.,OII//M„£ |a|<JV """

for some N sufficiently large.

We do not want to give the most general results. Instead of this we present a

result which is a typical application of the techniques developed here. It is the

counterpart of Lemma 1.5 and Corollary 1.6 in [2].

THEOREM 4.4. Let b E S^^q) n S^2cl(r',q;1) and a E S£l(r + l,q) D

S™ci(r' + ') 9! 7) be such that IsN, 1<<7<oo, r > n/q, r' <2r — n/q and m\ <

m,i+r' — r. Let 0 < p < q and m2 be such that n(max{l, 1/p} — 1) — 2r + n/q <

min{0, m2} and m2 < I. Define

Rr.= BoA- E ~Op(dpD^a).\a\<l

Then f E Hm>+m2+r-'>p n H™^m2+r'~l'p(l) implies RJ E Hr<p n H^Sil).

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354 JURGEN MARSCHALL

PROOF, (i) Let us first prove that

R, := B o Ay - E -.Op(d^bD^a)(9) \*\<i a-

ESr+m2-lif,9)nS^'crlir',q;1),

ml := max{mi + m2,roi + m'2,m'y + m2}. Ri E 5™1+m2_i(r,q) is plain by the

results of the preceding chapter. The proof that

So^i- E ^Op(dlbDyy)ES^'crl(r',q;1)\a\<l

is similar to Theorem 3.2. One has only to replace (2.25) by inequality (8). Hence,

(9) follows, if we can show that

(10) a2+a3ES^l(r',q;1).

Now there is c$ supported in |n| > igo(l + |£|) sucn that

[d°cs(r,)\ < (7(1 + |e|)-'(l + M)'-IQI and c€(_?)a(-, 0 = Oa(., 0 + o8(-, 0-

Hence, analogous to the second step of the proof of Theorem 4.1 one may prove

iic€(_?)a(-, o\\Hr', < c(i + icrir'iK, e)iiHr'+(,,.met mcl

But this yields (10).

(ii) Since m < I, r > n/q and 7712 + 2r — n/q > n(max{l, 1/p} — 1) the proof of

Theorem 4.1 shows that (A2 + A3)f E H™2t+r''p(7) and hence B o (A2 + A3)f E

Hmciil)- Finally, the results of the preceding chapter show that B o (A2 + A3)f E

HT'p.

This completes the proof of the theorem. D

The questions arises, whether we have

(11) Rt E S™>+m2-lir,q)nS™'crlif'i<i;i)-

By (9) this follows, if we can show Bo(A2 + A3) E S0rii+m2-l(r,q)nS^i(r',q;1).

Consider first B o A3.

LEMMA 4.5. Under the hypothesis of Theorem 4.4 we have

BoA3ES^+m2-'(r,q)nSZrl(r',q;1).

PROOF. Analogous to step (c) of the proof of Theorem 3.2 we get by summing

over all j such that 23 > 5 • (1 + |f|),

\obcas(x, 01 < E / l6(x' ̂ + ^(V, 0\dv

< C (J223{m2-{r+l-n/q))) \\a[\Hr+,,„

< (7(1 -I- \£\\"i\+m2-(r+l-n/q) ^

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 355

Further, for \n\ > 5 • (1 + |£|) we have

|9»afb(x,i + n)\< (7(1 + Mr*-I*l(1 + |£|)-W

K^6(-, e + r?)||//r.<, < (7(1 + |n|r2-l«l(l + iei)-|/J|.

Hence, since m2 < I, Theorem 2.2 yields

||ctBoA3(-,0II//- <C\\a3(;0\\H^2, <C(l + |£iri+m2-;'9.

Since a similar estimate holds for d^asoA3, we get B o A3 E S™1+m2~l(r, q). The

proof that B o A3 E 5™ci"~'(r', q) is completely analogous. □

Next consider B o A2 and suppose that r is finite. Then by (2.10) and (2.12) a

necessary condition for B o A2 E 5™1+m2~'(r, q) is m2 > 0. To get a little more

insight into the problem observe that crBOj42(.,0 = e~ix ^Op(b)(eiy'^a2(-, £))(x).

Then from the inequalities

||. • /ll//- < C([[g\[L~[\f[[H„, + ||<?||_^2||/|M, « > 0,

II.-/II*-. <«_,-. H/Hh.,,, s<0,c©,2

and from Theorem 2.2 it is easy to see that

(12) \\o-boaA; Oil//- < C(l + |e|)mi+mf''.

Hence, even if TO2 > 0, the difficulties appear by estimating d?<7Bo>42. We don't

know the answer to this problem. Thus, (11) remains open.

4.3 Microlocal ellipticity. A symbol o 6 Sol(r,q)nS™cl(r',q;~i) is called microlo-

cally elliptic iff for each (xo, £o) E 7 there exist a neighborhood U of _o and a conic

neighborhood c of £0 such that

(13) |a(_,oi><?(i+ieir

for each (x, 0 E U x c.

THEOREM 4.6. Let a E 50n(r,<7) nS™l(r',g;'() be microlocally elliptic and

suppose that m > 0,1 < q < 00, r > n/q, r' < 2r — n/q and 0 < p < q. Then

f E Hm+*'p and Af E Hr* n ^(7) imply f E HZY'^l)- □

Let us first develop some further tools needed in the proof of the theorem.

THEOREM 4.7. Let F: Rn x R —► R be smooth and suppose that 0 < p < 00,

r > n/p and r' < 2r - n/p. Then f E Hr'p n #^(7) implies F(-,f) E H[£ n

<cpih)-

PROOF. We may suppose that suppF c K x R for some compact set K c Rn.

Then one has F(-, f) = Af for some operator A E Sy(oo, 00). Moreover, a a satisfies

\d°dl<TA(x, 01 < C(l + |{|)--+»/«+l»|-|/»l

whenever |a| > r — n/q (see [15]). Hence proceeding as in Theorem 4.1 the result

follows. □

For similar results in this context let us mention [3, 15, 18, and 23].

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356 JURGEN MARSCHALL

LEMMA 4.8. LetbE 50m2 (r, q) n _£d(r', <?; 7) and a E 50mi (r, g) n sfy(r',q; 7)

6e such that m2 < 0,1 < q < 00, r' < 2r-n/q, r + n/q < r for some 0 < r < 1 and

max{mi + rri2,my + m2} <my+m2 + r'-r. Suppose that 0 < p < q is such that

n(max{l,l/p}-l)-2r+T+n/_ < m2. Then f E Hmt+m'+r>'nH%>l+m2+r'~T'p(<i)

implies (Bo A- Op(ba))f E Hr'p n Hrmf,(~t). □

For the proof of the lemma one may proceed as in Corollary 3.4 and then apply

Theorem 4.1. Note that the full strength of the theorem is needed.

PROOF OF THEOREM 4.6. We construct a microfocal paramatrix for A. Let

(xo, £0) E 7 and choose a test function tp supported in U and a symbol tp E SyQ

supported in the cone c. If tp and tp are chosen appropriately, it is not difficult to

see from (13) that

6(x,0 :=^(x)tP(f)a(x,f)-1ESom(r,q)nS-^(r',q;(xo,io)).

In fact, apply Theorem 4.7 with F(t) := t~x and use Corollary 4.3 and (8).

We then have

(14) v(x)tP(D)f = B(Af) -(Bo A- Op(ba))f.

Hence the lemma yields / E H™*lr+T'p((xo, tio)) for some 0 < r < 1. Now applying

the lemma repeatedly we get the conclusion. □

The results of this section have immediate applications to the microlocal regu-

larity of quasi-linear equations. Let

(15) P(x,Df) = FQ(x,f,...,Df}f,...)+ E Fa(x,f,...,D0f,...)Daf\a\=m

be a quasi-linear differential operator with smooth functions Fa: Rn x AT|0|<m_i R

—► R having compact support in the x-variable.

Let / E Hr+m'q, r > n/q and define a(x, 0 := E\a\=m Fa(x,..., D^f,... )f<*.

Define 7 := {(x, f) E R" x 5n_1: a(x, f) ^ 0} to be the set of points where P

and / are noncharacteristic. Observe that 7 is precisely the set of points where a

is microlocally elliptic. Note further that by Theorem 4.7

a E S?^ + l,q)C S?(r,q) n S™cl(r + l,g;i).

Hence repeated applications of Theorems 4.6 and 4.7 yield

THEOREM 4.9. Let P(x,Df) be a quasi-linear differential operator of order

m. Suppose that 1 < q < 00, r > n/q, f E Hr+m'q and P(x,Df) = 0. Then

f E Hr^~,n'q m,q("i) where 7 is the set of points where P and f are noncharacter-

istic. U

COROLLARY 4.10. Under the hypothesis of Theorem 4.9, if {x0} x 5n_i C 7

then f is smooth in a neighborhood of xq. □

In fact, there is a neighborhood U of xo such that U x 5n_1 C 7.

5. Coordinate transformations. Let E be a Banach space of bounded func-

tions defined on R" (i.e. let E --> L°°(Rn)). Denote by 5m(.E,r) the space of all

symbols such that for each multi-index a

(1) \d1a(x,0\<Ca(l + \f[)m-^.

(2) !l^a(.,OIU<C(l + |£|)m+T-|Q|.

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 357

For example, one has S™(r,q) = Sm(Hr'q,Sr). We shall prove that, under some

reasonable conditions on E, Sm (E, r) is invariant under regular diffeomorphisms

<p: Rn -» Rn, i.e. one has \da(p(x)[ < Ca for all a ^ 0 and c_1 < |det J^(x)| < c

for some constant c > 1 where J^ denotes the Jacobian of (p. Note that </> is regular

iff <p~l is regular. Define T$f(x) := \detJ^,(x)\f o <p(x). Then using a method of

Kuranishi we have for a E Sm(E,r)

(3) _V. o Op(a) o T*f(x) = JL; J j ei{x~y)%(x, y, 0/(2/) dy d£

where

(4) M*.V.O := \detJ^(x)\-1\detJ(x,y)\-1a((p-1(x),tJ(x,y)-10-

Here we have defined

J(x,y)-= J^-i(y + t(x-y))dt.Jo

For this and the following see, for example, [20, Chapter 1].

Using a Taylor expansion one gets a symbol a^ such that T^-i o Op(a) o T^f =

Op(al4>)f. In fact, a^ is given by

**(*,£):= E ^D^b<p(x,x,f) + fl(x,f),

(5) r'(_,0~7rVy -J\l-t)1-1

. j j j<.*-v)-l*-i)d%D$bi,(x, x + t(y- x),n) dydn dt.

THEOREM 5.1. Suppose that E is invariant under regular diffeomorphisms, i.e.

(6) HWb < CII/IU.Suppose further that

(7) \\9-f\\E<C\\f\\E

for each smooth g such that |daa(x)| < Ca for all multi-indices a. Then A E

Sm(E,r) implies T0-. o AoT^E Sm(E,r). □

COROLLARY 5.2. The symbol classes S™(r,q) are invariant under regular dif-

feomorphisms. O

PROOF OF THE THEOREM. The decomposition into elementary symbols re-

mains valid. It depends only on the fact that E is a Banach space. Hence, sup-

pose that a = Efclo2fcmMfc(x)t/'fc(0 where ||Mfc|U~ < C, \\Mk\\E < C2kr and

\dai>k(£)\ < C,2-fc|a| for \a\ < 4N.

It follows from (7) that (g, f) —► g ■ f is a continuous bilinear mapping. Hence,

there is a natural number A^ such that

(8) ||.-/IU<C( sup ||3a_||_~]||/||_:\>I<M) /

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358 JURGEN MARSCHALL

for all g and /. Now choose / > r and let N > N0 + 21. We shall prove that for all

|a| < N

0) |a^(x,oi<c(i + |ei)m-|a|.

(10) \\^(;0\\E<C(l + \f\r+T-M.

Since N may be chosen arbitrarily large, the theorem follows.

We haveOO

<(x, 0 = E lkmMk(r \x))(blk(x, 0 + r'k(x, 0)fc=0

where we have defined

6i(x,0:=|detJ4x)|-1 E \d?Dy {ldet J^^T' M^i^vTH)}\a\<ia' y=x

and

r*<*'fl:== (2^T E ±f Je«*-yH*>-Ud$DZclk(x,y,T,)dydr,,

clk(x,y,n):=\detJ^>(x)\-1 [ (1 - t)l~1\detJ(x,x+ t(y - x))^1Jo

■ tpk(lJ(x,x + t(y - x))-xn)dt.

Now, since <p is regular, it is easy to see that blk is supported in the annulus |£| ~ 2k

and that for \(3[ < N0 and |-yI < N

\d0xdJblk(x,f)\<C2-k^.

Hence, by (6) and (8) Ylk=o2kmM(<l>~1(x))bk(xi 0 satisfies (9) and (10). Finally

we have for \/3\ < N0, M < N, \p\ < 2N and |a| = I,

2km\d^yd^D-clk(x,y,n)[ < C2~kl(l + \v\)m~M.

Using standard techniques it follows that

2km\d^dJr{(x,0\<C2-kl(l + \f\r-^

(compare e.g. [20, Chapter 1]).

But then, since I > r, (9) and (10) follow easily. D

Appendix. We will prove Theorem 1.3 and Proposition 1.4.

PROOF OF THEOREM 1.3. (i) Let us first recall the proof for the case 1 < p <

oo given in [16, Chapter 7). It is easy to see that

[\f\pdx~ E IWkf[pdx

and hence, if s € N,

11/11*" ~( E II^/H«..p) •VfceZ" /

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PSEUDO-DIFFERENTIAL OPERATORS IN SOBOLEV SPACES 359

For arbitrary s > 0 the statement then follows by complex interpolation. This

approach can also be applied to the case p = oo. So let us concentrate on the case

0<p< 1.(ii) If 0<p< 1, then by (1.4)

(i) ii/iiw < c e UkfrH.,.fcez"

In order to prove the reverse inequality we show first

(2) E lhk/H„p<c||/H„p.fc_zn

Let tp E Cg° be supported in the unit ball with J tpdx = 1. Then we obtain

E Ukf[[vhP < C\\ sup sup bt*(Vfc/)|||_p-fci^n 0<t<ifcez"

Now suppose that / is an atom, i.e. suppose that / is support in a cube Q such that

||/||l°° < IQI-1'''' and that in case [Q[ < 1 the moment condition /xaf(x)dx = 0

for |a| < N :=* [n(l/p - 1)] holds (compare [7]). Then it suffices to prove

(3) || sup sup [tpt * (tpkf)\\Lp < C0<t<lfc_Z"

for each atom /, and (2) follows.

(iii) Let Q = Q(x0, d). If |<3| > 1, then sup0<t<i supfc€Zn |<£>t*(V>fc/)l is supported

in Q(x0,d+ 1) and hence, (3) follows from ||/||l» < \Q\~1/p-

In case |Q| < 1 let Q* be Q doubled, i.e. let Q* = Q(x0,2d). Then ||/||l~ <|Q|-i/p implies

(4) / sup sup \tpt * (tpkf)\pdx < C.JQ* 0<t<lfc€Z"

For x £ Q* we make a Taylor expansion

<Pt(x - y)tpk(y) = E —^yiPtix-y^kiy^^iy-xoT + RN-\a\<N

Because of the moment condition we have

<Pt * (tpkf)(x) = I RNf(y)dy.

Since tpt is supported in a ball |x| < t, we have for x ^ Q* and y E Q, \Rn\ <

(7|x - Xo\~N~n~1\y - x0\N+1 which implies

\<Pt * Wkf){x)\ < C||/||l»|x - xol-^-""1 f \y- x0\N+1dyJq

< cdN+n+i-n/p\x - _or"-n-1.

Since (N + n + l)p > n, we obtain

(5) / sup sup [tpt * (tpkf)\pdx < CJR"\Q' 0<t<lkeZn

which together with (4) yields (3). Thus (2) is proved.

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360 JURGEN MARSCHALL

(iv) Similarly to (2) one may prove for s E N

(6) E ll^/ll*..p<c||/H5...p.fc€Z"

For the remaining values of s > 0 we use the complex interpolation method devel-

oped in [21, Chapter 2.4].

Let s = ei, 0 < 6 < 1 and I e N. We can find ip'k E S such that tp'k E C^ and

(7) \\(^k-rP'k)9\\PH,.P<(l + [k\)-n-1[[9[[pH>„p

for each 0 < X < I and g E Hx'p. Then with Triebel's notation let f(z) be 5'-

analytic such that / = /(-,8).

Note that tp'kf(z) is 5'-analytic, too. Now by the first step of the proof of [21,

Theorem 2.4.7] we get

II^/|IS,..p < [j^ jjtP'kf(-,it)[\pHO,ppo(Q,t)dt}

X (^jjtP'kf(;l + it)[[PHliPpy(Q,t)dt^j

with some positive kernels po and pi such that

-j-!-q j /*o(e, t)dt = | j py (6, t)dt = 1.

But then using a1~ebe < (1 - 6)a + 66 it follows from (6) and (7)

( E ll^/llSr-p) <cfsup||/(.,JOII//o,p + sup||/(.,l + zr)||B1,p).

Now f(z) can be chosen in such a way that the right-hand side is dominated by

C||/||//»p- Hence, (6) holds for arbitrary s > 0. □

PROOF OF PROPOSITION 1.4. The case s = 0 of the proposition is an easy con-

sequence of Holder's inequality and in case p = oo of the John-Nirenberg inequality

[10]. When s ^ 0 choose a test function tp' such that tp' = 1 on a neighborhood of

suppi/;. Then for / € Ha'p we have with tp'k(x) := tp'(x — fc)

Js(^Pkf) = tP'kJs(tPkf) + (1 - tP'k)Js(tPkf).

By the case s = 0, tp'kJa(tpkf) E hq. Moreover, g —* (l — tp'k)Ja(tpkg) is a smoothing

operator, and hence, (1 - tp'k)Js(tpkf) E S. Thus we obtain tpkf E Hs'q. Finally,

the estimates' independence of fc 6 Z" is a consequence of the translation invariance

of the __ s'9-quasi norm. □

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Universitat der Bundeswehr Munchen, Fakultat fur Informatik, Institut

fur Mathematik, Werner-Heisenberg-Weg 39, D-8014 Neubiberg, Federal Re-public of Germany

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