A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant...

Integr. Equ. Oper. Theory 76 (2013), 1–23

DOI 10.1007/s00020-013-2052-6

Published online April 4, 2013c© Springer Basel 2013

Integral Equationsand Operator Theory

A Family of Fundamental Solutions ofElliptic Partial Differential Operatorswith Real Constant Coefficients

Matteo Dalla Riva

Abstract. We present a construction of a family of fundamental solutionsfor elliptic partial differential operators with real constant coefficients.The elements of such a family are expressed by means of jointly realanalytic functions of the coefficients of the operators and of the spatialvariable. The aim is to write detailed expressions for such functions.Such expressions are then exploited to prove regularity properties inthe frame of Schauder spaces and jump properties of the correspondingsingle layer potentials.

Mathematics Subject Classification (2010). 35E05, 35B20.

Keywords. Fundamental solutions, elliptic partial differential operatorswith real constant coefficients, layer potentials.

1. Introduction

We fix once for all

n, k ∈ N, n ≥ 2, k ≥ 1.

Here N denotes the set of natural numbers including 0. Then we denoteby N(2k, n) the set of all multi-indexes α ≡ (α1, . . . , αn) ∈ N

n with |α| ≡α1 + · · · + αn ≤ 2k. We denote by R(2k, n) the set of the functions a ≡(aα)α∈N(2k,n) from N(2k, n) to R. We note that R(2k, n) can be identifiedwith a finite dimensional real vector space. Accordingly we endow R(2k, n)

This work was supported by the Portuguese Foundation for Science and Technol-ogy (“FCT–Fundacao para a Ciencia e a Tecnologia”) via the post-doctoral grantSFRH/BPD/64437/2009 and by FEDER funds through COMPETE—Operational Pro-gramme Factors of Competitiveness (“Programa Operacional Factores de Competitivid-ade”) and by Portuguese funds through the Center for Research and Development inMathematics and Applications (University of Aveiro) and the Portuguese Foundation for

Science and Technology (“FCT–Fundacao para a Ciencia e a Tecnologia”), within projectPEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.

2 M. Dalla Riva IEOT

with the corresponding Euclidean norm |a| ≡ (∑

α∈N(2k,n) a2α)1/2. Then we

set

ER(2k, n)≡{

a ≡ (aα)α∈N(2k,n) ∈ R(2k, n) :∑

|α|=2k

aαξα �= 0 ∀ξ ∈ Rn\{0}

}

.

We note that ER(2k, n) is an open non-empty subset of R(2k, n). Then, foreach a ∈ ER(2k, n) we denote by L[a] the partial differential operator definedby

L[a] ≡∑

α∈N(2k,n)

aα∂α1x1

. . . ∂αnxn

(see also Sect. 2 below). So that, L[a] is a real constant coefficients ellipticpartial differential operator on R

n of order 2k.In this paper we show a construction of a function S such that

S is a real analytic function from ER(2k, n) × (Rn\{0}) to R ; (1)S(a, ·) is a fundamental solution of L[a] for all a ∈ ER(2k, n). (2)

Condition (2) means that S(a, ·) defines a locally integrable function on Rn

such that L[a]S(a, ·) = δ0 in the sense of distributions on Rn, where δ0

denotes the delta Dirac distribution with mass at 0.The aim is to write a suitable detailed expression for S and to exploit it

to prove regularity properties in the frame of Schauder spaces and jump prop-erties of the single layer potential corresponding to the fundamental solutionS(a, ·).

In Theorem 5.1 below we introduce a function S which satisfies theconditions in (1), (2) and in Theorem 5.2 we write a detailed expressionfor S. In particular, we show that there exist a real analytic function Afrom ER(2k, n) × ∂Bn × R to R, and real analytic functions B and C fromER(2k, n) × R

n to R such that

S(a, x) = |x|2k−nA(a, x/|x|, |x|) + log |x|B(a, x) + C(a, x) (3)

for all (a, x) ∈ ER(2k, n) × (Rn\{0}), where B and C are identically 0 if thedimension n is odd. The functions A and B play an important role when weconsider the regularity and jump properties of the single layer potential cor-responding to S(a, ·). Therefore, we investigate the power series expansion ofA(a, θ, r) with respect to the “radius” variable r and of B(a, x) with respectto the spatial variable x (cf. Theorem 5.2 below). Then we denote by L0[a]the principal term of L[a]. Namely, we set

L0[a] ≡∑

|α|=2k

aα∂α1x1

. . . ∂αnxn

(4)

for all a ∈ ER(2k, n) (see also Sect. 2 below). In Theorem 5.3 we show thatthere exists a real analytic function S0 from ER(2k, n) × (Rn\{0}) to R suchthat S0(a, ·) is a fundamental solution of L0[a] for all a ∈ ER(2k, n). We alsoprovide an expression for S0 in terms of the coefficients of the power seriesexpansions of the functions A and B which appear in (3).

Vol. 76 (2013) A Family of Fundamental Solutions 3

Then we turn to consider the single layer potential corresponding to thefundamental solution S(a, ·). We fix once for all

m ∈ N\{0} and λ ∈]0, 1[.

We fix a set Ω such that

Ω is an open bounded subset of Rn of class Cm,λ.

For the definition of functions and sets of the usual Schauder class Ch,λ,with h ∈ N, we refer for example to Gilbarg and Trudinger [13, § 4.1 and § 6.2].We denote by v[a, μ] the single layer potential with density μ ∈ Cm−1,λ(∂Ω)corresponding to the fundamental solution S(a, ·), a ∈ ER(2k, n). Namely,v[a, μ] is the function from R

n to R defined by

v[a, μ](x) ≡∫

∂Ω

S(a, x − y)μ(y) dσy ∀x ∈ Rn, (5)

for all a ∈ ER(2k, n) and all μ ∈ Cm−1,λ(∂Ω), where ∂Ω denotes the bound-ary of Ω and dσ denotes the area element on ∂Ω. In Theorem 6.1 we showthat v[a, μ] is a function of class C2k−2 on R

n. In Theorem 6.6 we show thatthe restriction of v[a, μ] to the closure of Ω is a function of class Cm+2k−2,λ

and the restriction of v[a, μ] to Rn\Ω belongs to Cm+2k−2,λ in a local sense

which will be clarified. However, the derivatives of order 2k − 1 of v[a, μ](x)with respect to the spatial variable x are not continuous on R

n and displaya jump across the boundary of Ω. In Theorem 6.8 we describe such a jumpproperty.

We observe that the result which we state in Theorem 5.1 resemblesthose which were proved by Treves in [27] and Mantlik in [20,21]. In theirworks, Treves and Mantlik consider a family of hypoelliptic complex par-tial differential operators {L(ζ)}ζ∈Z with coefficients which are holomorphicfunctions of a complex parameter ζ in a Stein manifold M. Provided thatthe strength of the operators L(ζ) is constant for all ζ ∈ M, Mantlik showsthe existence of an holomorphic map S from M to a suitable space of distri-butions such that S(ζ) is a fundamental solution of L(ζ) for all ζ ∈ M. Theresult of Mantlik extends the previous one of Treves, where the existence ofthe function S was proved only locally in a neighborhood of a fixed point ofM. The hypoellipticity and equal strength conditions required by Treves andMantlik for the operators are weaker conditions than the ellipticity requiredin this paper and the result of Mantlik can be exploited to prove the existenceof a function which satisfies the conditions in (1) and (2).

However, our Theorems 5.1 is not obtained as a corollary of the resultsof Treves and Mantlik. Indeed, we adopt here a different approach based onthe techniques developed by John in [14, Chapter III] for the construction ofa fundamental solution of a partial differential operator with analytic coef-ficients. Such an approach leads to an explicit construction of a function Sas in (1), (2) which in addition can be described as in Theorem 5.2 [see alsoequality (3)]. We exploit such a description in order to develop a suitablepotential theory in the framework of Schauder spaces.


The principal application which motivates our approach is the creationof a useful tool in order to study perturbed boundary value problems bymeans of a potential theoretic method. In view of such an application, it isimportant to understand the dependence of layer potentials corresponding toa certain fundamental solution of a partial differential operator upon pertur-bations of the support of integration and of data such as the coefficients ofthe operator and the density function. We mention for example the differen-tiability results for the dependence of layer potentials upon deformations ofthe support of integration obtained by Potthast in the framework of Schau-der spaces (cf., e.g., [24–26]) and by Costabel and Le Louer in the frameworkof Sobolev spaces (cf., e.g., [5,6]). Such results are then exploited to studyperturbations in scattering problems.

Also, Lanza de Cristoforis and collaborators have developed a methodbased on potential theory with the aim of proving real analyticity results inthe framework of Schauder spaces for the dependence of the solutions of cer-tain elliptic boundary value problems upon perturbations of the domain ofdefinition and of the data (see, e.g., [15,16]). In order to apply such a method,one first has to verify the real analytic dependence of the layer potentials onboth variation of the support of integration and on data. In [18,19] Lanza deCristoforis and Rossi have considered the layer potentials associated to theLaplace and Helmholtz operators. In [9] the case of layer potentials corre-sponding to the second order complex constant coefficient elliptic operatorshas been investigated, and in [17] Lanza de Cristoforis and Musolino havestudied the periodic analog (see also [7] for layer potentials corresponding tothe Lame equations and to some higher order operators).

We observe that both in [9,17] the existence of suitable families of fun-damental solutions is postulated in order to show real analyticity results forthe layer potentials. In the case of operators of order two, the family of fun-damental solutions constructed in this paper is of the type considered in [9].Moreover, in order to treat different classes of operators, for example oper-ators of order higher than two, one needs suitable families of fundamentalsolutions. In this sense, the present paper can be considered as a first steptoward the extension of the approach of Lanza de Cristoforis and collabo-rators to the case of general elliptic partial differential operators with realconstant coefficients. Also, the construction presented here can be exploitedto produce analog constructions for partial differential operators with com-plex or quaternion constant coefficients (cf. [10]) and for particular systemsof differential operators (cf. [8]).

The paper is organized as follows. In Sect. 2 we introduce some standardnotation and we recall a classical result on real analytic functions. Sections3 and 4 are devoted to analysis of some auxiliary functions and are thereforerather technical. In Sect. 3 we consider a function v which is solution of theequation L[a]v = 1 and which vanish together with its derivatives of ordersmaller than or equal to 2k − 1 on a hyperplane of R

n. In Sect. 4, we intro-duce the auxiliary functions W0, W1, and W2. We show that one can exploitW0, W1, and W2 to define a distribution Sa which is a fundamental solutionof the operator L[a] (cf. Proposition 4.1). In Sect. 5 we are ready to prove


Theorems 5.1, 5.2, and 5.3 where we introduce and study our functions Sand S0. We observe that in Theorem 5.1 we verify that the distribution Sa ofSect. 4 coincides with the distribution defined by the function S(a, ·). In thelast Sect. 6, we consider the single layer potential v[a, μ] and we prove The-orems 6.1, 6.6, and 6.8 where we investigate regularity and jump propertiesof v[a, μ].

2. Some Notation and Preliminaries

For standard definitions of calculus in normed spaces, we refer, e.g., to Cartan[2]. We understand that a finite product of normed spaces is equipped withthe sup-norm of the norm of the components, while we use the Euclideannorm for R

n. For all x ∈ Rn, xj denotes the j-th coordinate of x, |x| denotes

the Euclidean modulus of x, and Bn denotes the unit ball {x ∈ Rn : |x| < 1}.

A dot ‘·’ denotes the inner product in Rn. If X is a subset of R

n, then clXdenotes the closure of X and ∂X denotes the boundary of X . If O is anopen subset of R

n, and f is a function from O to R, and x ∈ O, then thepartial derivative of f with respect to xj at x is denoted by ∂xj

f(x). Then∂α

x f(x) ≡ ∂α1x1

. . . ∂αnxn

f(x) for all multi-index α ≡ (α1, . . . , αn) ∈ Nn and

∂xf(x) denotes the column vector (∂x1f(x), . . . , ∂xnf(x)). If h ∈ N, then the

space of the h times continuously differentiable real-valued functions on Ois denoted by Cm(O). The subspace of Ch(O) of those functions f whosederivatives ∂α

x f of order |α| ≤ h can be extended to a continuous functionon clO is denoted Ch(clO). The space of n × n real matrices is denoted byMn(R). If M ∈ Mn(R) then M t denotes the transpose matrix of M . For thedefinition and properties of analytic operators, we refer, e.g., to Deimling [11,p. 150]. In the sequel we shall need the following classical lemma.

Lemma 2.1. Let h1, h2 ∈ N\{0}. Let X ⊆ Rh1 , Y ⊆ R

h2 . Assume that Y iscompact. Let τ be a finite measure on the measurable subsets of Y. Let f bea real analytic function from X × Y to R. Let F be the function from X toR defined by F (x) ≡ ∫

Y f(x, y) dτy for all x ∈ X . Then F is real analytic.

Here, we understand that a function f defined on a subset X of a Banachspace is real analytic if f is the restriction to X of a real analytic functiondefined on an open neighborhood of X .

3. The Auxiliary Function v

As a first step in the construction of our real analytic function S as in (1), (2),we will show in this section the existence and uniqueness of a real analyticfunction v from ER(2k, n)×R

n ×∂Bn ×R to R such that L[a]v(a, x, ξ, t) = 1for all (a, x, ξ, t) ∈ ER(2k, n) × R

n × ∂Bn × R and that ∂αx v(a, x, ξ, t) = 0 for

all (a, x, ξ, t) ∈ ER(2k, n) × Rn × ∂Bn × R with x · ξ = t and all α ∈ N

n with|α| ≤ 2k − 1. Then we will investigate some properties of such a function v.


We introduce the following notation. If a ≡ (aα)α∈N(2k,n) ∈ ER(2k, n),then we set

P [a](ξ) = P [a](ξ1, . . . , ξn) ≡ ∑

α=(α1,...,αn)∈N(2k,n)

aαξα11 . . . ξαn

n ,

P0[a](ξ) = P0[a](ξ1, . . . , ξn) ≡ ∑

α=(α1,...,αn)∈Nn, |α|=2k

aαξα11 . . . ξαn

n .

So that, P [a] is a real polynomial in n variables of degree 2k and P0[a] is thehomogeneous term of P [a] of degree 2k. Then we have the following lemma.

Lemma 3.1. Let a ≡ (aα)α∈N(2k,n) ∈ ER(2k, n). Let ρ ∈ R be such that

ρ ≥ 1 +

∑|α|≤2k−1 |aα|

infξ∈∂Bn|P0[a](ξ)| . (6)

Let Dρ ≡ {z ∈ C : |z| < ρ}. Let va be the function from Rn × ∂Bn × R to R

defined by

va(x, ξ, t) ≡ 12πi

∫

∂Dρ

e(x·ξ−t)ζ

ζP [a](ζξ)dζ ∀(x, ξ, t) ∈ R

n × ∂Bn × R, (7)

where i denotes the imaginary unit, namely i2 = −1, and where ζξ denotesthe complex vector (ζξ1, . . . , ζξn). We understand that the line integral in (7)is taken over the parametrization ρeis, s ∈ [0, 2π[.

Then va is the unique real analytic function from Rn × ∂Bn × R to

R such that L[a]va(x, ξ, t) =∑

|α|≤2k aα∂αx va(x, ξ, t) = 1 for all (x, ξ, t) ∈

Rn × ∂Bn × R and that ∂α

x va(x, ξ, t) = 0 for all (x, ξ, t) ∈ Rn × ∂Bn × R with

x · ξ = t and all α ∈ Nn with |α| ≤ 2k − 1.

Proof. By the membership of a in ER(2k, n), by condition (6), and by astraightforward calculation one verifies that the polynomial P [a](ζξ) has nocomplex zeros ζ outside of the open disk Dρ, for all fixed ξ ∈ ∂Bn. ThenLemma 2.1 and standard properties of real analytic functions imply that thefunction va defined by (7) is real analytic from R

n×∂Bn×R to R. By equalityL[a]e(x·ξ−t)ζ =

∑|α|≤2k aα∂α

x e(x·ξ−t)ζ = P [a](ζξ), by standard theorems ondifferentiation under integral sign, and by the Cauchy integral formula onededuces that

L[a]va(x, ξ, t) =1

2πi

∫

∂Dρ

L[a](e(x·ξ−t)ζ

)

ζP [a](ζξ)dζ =

12πi

∫

∂Dρ

1ζ

dζ = 1 (8)

for all (x, ξ, t) ∈ Rn ×∂Bn ×R. By equality e(x·ξ−t)ζ =

∑∞j=0 (x · ξ − t)jζj/j!

and by standard theorems on summation under integral sign one has

va(x, ξ, t) =∞∑

j=0

aj(ξ)j!

(x · ξ − t)j ∀(x, ξ, t) ∈ Rn × ∂Bn × R (9)

with

aj(ξ) ≡ 12πi

∫

∂Dρ

ζj−1

P [a](ζξ)dζ ∀ξ ∈ ∂Bn, j ∈ N. (10)


Then, by a standard argument based on the Cauchy integral formula one ver-ifies that

aj(ξ) ∈ R ∀ξ ∈ ∂Bn, j ∈ N (11)

and that

aj = 0 ∀j ∈ {0, 1, . . . , 2k − 1}, a2k(ξ) = 1/P0[a](ξ) ∀ξ ∈ ∂Bn. (12)

Now, by (8), (9), and (12), and by standard calculus in Banach space oneverifies that L[a]va(x, ξ, t) = 1 for all (x, ξ, t) ∈ R

n × ∂Bn × R and that∂α

x va(x, ξ, t) = 0 for all (x, ξ, t) ∈ Rn × ∂Bn × R with x · ξ = t and all α ∈ N

n

with |α| ≤ 2k − 1. The uniqueness of the function va is a consequence ofCauchy–Kovalevskaya Theorem. �

We are now ready to show in the following Proposition 3.2 the existenceand uniqueness of the auxiliary function v.

Proposition 3.2. There exist a unique real analytic function v from ER(2k, n)×R

n × ∂Bn × R to R such that L[a]v(a, x, ξ, t) = 1 for all (a, x, ξ, t) ∈ER(2k, n) × R

n × ∂Bn × R and that ∂αx v(a, x, ξ, t) = 0 for all (a, x, ξ, t) ∈

ER(2k, n) × Rn × ∂Bn × R with x · ξ = t and all α ∈ N

n with |α| ≤ 2k − 1.

Proof. Let ER,l(2k, n) denote the open subset of ER(2k, n) defined by

ER,l(2k, n) ≡{

a ∈ ER(2k, n) :∑

|α|≤2k−1

|aα| < l and infξ∈∂Bn

|P0[a](ξ)| >1l

}

for all l ∈ N\{0}. Then ER,l(2k, n) ⊆ ER,l+1(2k, n) for all l ∈ N\{0} and∪∞

l=1ER,l(2k, n) = ER(2k, n). Let now l ∈ N\{0} be fixed. Let ρl ≡ 1 + l2 andDρl

≡ {z ∈ C : |z| < ρl}. Let vl denote the function from ER,l(2k, n) × Rn ×

∂Bn × R to R defined by

vl(a, x, ξ, t) ≡ 12πi

∫

∂Dρl

e(x·ξ−t)ζ

ζP [a](ζξ)dζ (13)

for all (a, x, ξ, t) ∈ ER,l(2k, n) × Rn × ∂Bn × R. One verifies that ρl ≥ 1 +∑

|α|≤2k−1 |aα|/ infξ∈∂BnP0[a](ξ) for all a ∈ ER,l(2k, n). Then P [a](ζξ) �= 0

for all ζ ∈ C with |ζ| = ρl and all (a, ξ) ∈ ER,l(2k, n) × ∂Bn (see also theproof of Lemma 3.1). Thus, Lemma 2.1 implies that vl is real analytic fromER,l(2k, n)× R

n × ∂Bn × R to R. Moreover, Lemma 3.1 implies that vl is theunique real analytic function from ER,l(2k, n) × R

n × ∂Bn × R to R such thatL[a]vl(a, x, ξ, t) = 1 for all (a, x, ξ, t) ∈ ER,l(2k, n) × R

n × ∂Bn × R and suchthat ∂α

x vl(a, x, ξ, t) = 0 for all (a, x, ξ, t) ∈ ER,l(2k, n) × Rn × ∂Bn × R with

x·ξ = t and all α ∈ Nn with |α| ≤ 2k−1. Hence vl(a, x, ξ, t) = vl′(a, x, ξ, t) for

all (a, x, ξ, t) ∈ ER,l(2k, n)×Rn×∂Bn×R and all l, l′ ∈ N\{0} such that l ≤ l′.

Thus, one can define the function v from ER(2k, n) × Rn × ∂Bn × R to R by

setting v(a, x, ξ, t) ≡ vl(a, x, ξ, t) for all (a, x, ξ, t) ∈ ER,l(2k, n)×Rn×∂Bn×R

and all l ∈ N\{0}. Then v satisfies the conditions in the statement. �

In the following Propositions 3.3 and 3.4 we stress some further proper-ties of the auxiliary function v.


Proposition 3.3. Let v be the function from ER(2k, n) × Rn × ∂Bn × R to

R of Proposition 3.2. Then there exists a sequence {aj}j∈N of real analyticfunctions from ER(2k, n) × ∂Bn to R such that

v(a, x, ξ, t)=∞∑

j=0

aj(a, ξ)j!

(x · ξ − t)j ∀(a, x, ξ, t) ∈ ER(2k, n)×Rn×∂Bn × R,

where the series converges absolutely and uniformly in the compact subsets ofER(2k, n) × R

n × ∂Bn × R. Moreover, aj = 0 for j ≤ 2k − 1 and a2k(a, ξ) =1/P0[a](ξ) for all (a, ξ) ∈ ER(2k, n) × ∂Bn.

Proof. With the notation introduced in the proof of Proposition 3.2, let

aj,l(a, ξ) ≡ 12πi

∫

∂Dρl

ζj−1

P [a](ζξ)dζ ∀(a, ξ) ∈ ER,l(2k, n) × ∂Bn, (14)

for all j, l ∈ N, l ≥ 1. Then aj,l(a, ξ) ∈ R for all (a, ξ) ∈ ER,l(2k, n) × ∂Bn

and all j, l ∈ N, l ≥ 1 [cf. (10) and (11)]. Moreover, by arguing so as inthe proof of Proposition 3.2 for vl one verifies that the functions aj,l are realanalytic from ER,l(2k, n)×∂Bn to R for all j, l ∈ N, l ≥ 1. By definition (14)and by a straightforward calculation one verifies that

|aj,l(a, ξ)| ≤ l(1 + l2)j+1−2k ∀(a, ξ) ∈ ER,l(2k, n) × ∂Bn (15)

for all j, l ∈ N, l ≥ 1. Hence, by (13), by inequality (15), and by a straight-forward calculation one can prove that

vl(a, x, ξ, t) =∞∑

j=0

aj,l(a, ξ)j!

(x · ξ − t)j ∀(a, x, ξ, t)∈ER,l(2k, n)

×Rn×∂Bn×R,

for all l ∈ N\{0}, where the series converges absolutely and uniformly in thecompact subsets of ER,l(2k, n)×R

n×∂Bn×R. Since vl(a, x, ξ, t) = vl′(a, x, ξ, t)for all (a, x, ξ, t) ∈ ER,l(2k, n)×R

n ×∂Bn×R and all l, l′ ∈ N\{0} with l ≤ l′,one deduces that aj,l(a, ξ) = aj,l′(a, ξ) for all (a, ξ) ∈ ER,l(2k, n)×∂Bn and allj, l, l′ ∈ N with 1 ≤ l ≤ l′ (cf. proof of Proposition 3.2). Hence, one can definethe function aj from ER(2k, n) × ∂Bn to R by setting aj(a, ξ) ≡ aj,l(a, ξ) forall (a, ξ) ∈ ER,l(2k, n) × ∂Bn and all l ∈ N\{0}. Then {aj}j∈N satisfies theconditions in the statement [see also (12)]. �

Proposition 3.4. Let v be the function from ER(2k, n) × Rn × ∂Bn × R to R

of Proposition 3.2. Then there exists a unique real analytic function w fromER(2k, n) × ∂Bn × R to R such that v(a, x, ξ, t) = (x · ξ − t)2kw(a, ξ, x · ξ − t)for all (a, x, ξ, t) ∈ ER(2k, n) × R

n × ∂Bn × R.

Proof. It is a consequence of Proposition 3.3, of standard properties of realanalytic functions, and of the definition of v (see also Proposition 3.2). �


4. The Auxiliary Functions W0, W1, and W2

Let W0, W1, and W2 denote the functions from ER(2k, n) × (Rn\{0}) to R

defined by

W0(a, x) ≡ 14(2πi)n−pn

∫

∂Bn

x·ξ∫

0

v(a, x, ξ, t) sgn t dt dσξ, (16)

W1(a, x) ≡ − 1(2πi)n−pn

∫

∂Bn

v(a, x, ξ, 0) log |x · ξ| dσξ, (17)

W2(a, x) ≡ − 1(2πi)n−pn

∫

∂Bn

x·ξ∫

0

v(a, x, ξ, t) − v(a, x, ξ, 0)t

dt dσξ, (18)

for all x ∈ Rn\{0} and all a ∈ ER(2k, n), where pn ≡ 1 if n is odd and pn ≡ 0

if n is even and where v is the function of Proposition 3.2.In the following Proposition 4.1 we construct by means of the functions

W0, W1, and W2 a particular fundamental solution Sa for the partial differ-ential operator L[a], a ∈ ER(2k, n). The validity of Proposition 4.1 follows bythe results of John in [14, Chap. 3]. For the sake of completeness we includehere a proof.

Proposition 4.1. Let Sa be the distribution on Rn defined by

Sa ≡ pnΔ(n+1)/2W0(a, ·) + (1 − pn)Δn/2(W1(a, ·) + W2(a, ·)). (19)

Then Sa is a fundamental solution of the operator L[a].

Proof. Let a ∈ ER(2k, n) be fixed. Note that the functions W0(a, ·), W1(a, ·),and W2(a, ·) extend to continuous functions on the whole of R

n. Accordinglythe expression in the right hand side of (19) defines a distribution on R

n.Now assume that the dimension n is odd, so that pn = 1. By standard

theorems on differentiation under integral sign and by the definition of v oneverifies that

∂αx W0(a, x) =

14(2πi)n−1

∫

∂Bn

x·ξ∫

0

∂αx v(a, x, ξ, t) sgn t dt dσξ ∀x ∈ R

n\{0},

for all α ∈ Nn with |α| ≤ 2k (see also Proposition 3.2). Note that the func-

tion ∂αx W0(a, ·) extends to a continuous function on the whole of R

n for allα ∈ N

n with |α| ≤ 2k. Moreover, one has

L[a]W0(a, x) =1

4(2πi)n−1

∫

∂Bn

|x · ξ| dσξ =(−1)(n−1)/2

2nπ(n−1)/2((n − 1)/2)!|x| (20)

for all x ∈ Rn\{0} (cf. John [14, p. 9]). Thus L[a]W0(a, ·) and the function in

the right hand side of (20) define the same distribution on Rn. Hence, the dis-

tribution L[a]W0(a, ·) is a fundamental solution of Δ(n+1)/2 (see, e.g., John[14, p. 44]). It follows that L[a](Δ(n+1)/2W0(a, ·))=Δ(n+1)/2(L[a]W0(a, ·))=


δ0 in the sense of distributions in Rn. Thus Δ(n+1)/2W0(a, ·) is a fundamental

solution of L[a] and the proposition is verified for n odd.Now assume that the dimension n is even and pn = 0. Denote by W3

the function from Rn\{0} to R defined by

W3(a, x) ≡ − 1(2πi)n

∫

∂Bn

x·ξ∫

0

v(a, x, ξ, t) t log |t| dt dσξ ∀x ∈ Rn\{0}.

Then by the classical theorem on differentiation under integral sign and bythe definition of v one verifies that

∂αx W3(a, x)≡− 1

(2πi)n

∫

∂Bn

x·ξ∫

0

∂αx v(a, x, ξ, t) t log |t| dt dσξ ∀x∈R

n\{0} (21)

for all α ∈ Nn with |α| ≤ 2k (see also Proposition 3.2). Note that the func-

tion ∂αx W3(a, ·) extends to a continuous function on the whole of R

n for allα ∈ N

n with |α| ≤ 2k. Moreover, a straightforward calculation shows that

L[a]W3(a, x)

= − 12(2πi)n

∫

∂Bn

(x · ξ)2 log |x · ξ| dσξ +1

4(2πi)n

∫

∂Bn

(x · ξ)2 dσξ

=(−1)(n/2)−1

2n+1πn/2(n/2)!|x|2 log |x| + cn|x|2 ∀x ∈ R

n\{0}, (22)

where cn is a real constant which depends only on n (cf. John [14, p. 9]).Thus L[a]W3(a, ·) and the function in the right hand side of (22) definethe same distribution on R

n. Hence the distribution L[a]W3(a, ·) is a fun-damental solution of Δ(n/2)+1 (see, e.g., John [14, p. 44], also note thatΔ(n/2)+1|x|2 = 0). Now, by Proposition 3.3 one verifies that ∂α

x v(a, x, ξ, t) =(−1)|α|ξα∂

|α|t v(a, x, ξ, t) for all (x, ξ, t) ∈ R

n × ∂Bn × R, α ∈ Nn. Hence,

equality (21) and a straightforward calculation imply that

ΔW3(a, x) = − 1(2πi)n

∫

∂Bn

x·ξ∫

0

(∂2t v(a, x, ξ, t)) t log |t| dt dσξ (23)

for all x ∈ Rn\{0}. Then, by integrating by parts the integral in the right

hand side of (23) one deduces that

ΔW3(a, x) = − 1(2πi)n

∫

∂Bn

v(a, x, ξ, 0) log |x · ξ| dσξ

− 1(2πi)n

∫

∂Bn

x·ξ∫

0

v(a, x, ξ, t) − v(a, x, ξ, 0)t

dt dσξ

− 1(2πi)n

∫

∂Bn

v(a, x, ξ, 0) dσξ ∀x ∈ Rn\{0}.


So that

W1(a, x) + W2(a, x) = ΔW3(a, x) +1

(2πi)n

∫

∂Bn

v(a, x, ξ, 0) dσξ

for all x ∈ Rn\{0}. Then one observes that

L[a]

⎛

⎝ 1(2πi)n

∫

∂Bn

v(a, x, ξ, 0) dσξ

⎞

⎠ =sn

(2πi)n∀x ∈ R

n,

where sn denotes the n−1 dimensional measure of ∂Bn. Hence, L[a](W1(a, ·)+W2(a, ·)) differs from L[a](ΔW3(a, ·)) by a constant function. It follows thatL[a]Δn/2(W1(a, ·)+W2(a, ·)) = Δ(n/2)+1L[a]W3(a, ·) = δ0 in the sense of dis-tributions in R

n. Thus Δn/2(W1(a, ·) + W2(a, ·)) is a fundamental solutionof L[a] and the proposition is verified also for n even. �

In the following Proposition 4.2 we investigate some further propertiesof the functions W0, W1, and W2.

Proposition 4.2. There exist two real analytic functions A� and A fromER(2k, n) × ∂Bn × R to R, and two real analytic functions B and C fromER(2k, n) × R

n to R such that

W0(a, x) = |x|2k+1A�(a, x/|x|, |x|),W1(a, x) = |x|2kA(a, x/|x|, |x|) + log |x|B(a, x),

W2(a, x) = C(a, x), ∀(a, x) ∈ ER(2k, n) × (Rn\{0}).

Also, ∂αx B(a, 0) = 0 for all a ∈ ER(2k, n) and all α ∈ N

n with |α| ≤ 2k − 1.

Proof. Let A� be the function from ER(2k, n) × ∂Bn × R to R defined by

A�(a, θ, r)

≡ 14(2πi)n−pn

∫

∂Bn

θ·ξ∫

0

(θ · ξ − s)2kw (a, ξ, r (θ · ξ − s)) sgn s ds dσξ

(24)

for all (a, θ, r) ∈ ER(2k, n) × ∂Bn × R, where w is as in Proposition 3.4.Then, by definition (16), by equality v(a, x, ξ, t) = (x · ξ − t)2kw(a, ξ, x ·ξ − t), and by a straightforward calculation one verifies that W0(a, x) =|x|2k+1A�(a, x/|x|, |x|) for all (a, x) ∈ ER(2k, n) × (Rn\{0}). We now showthat A� is real analytic from ER(2k, n) × ∂Bn × R to R. By changing variableof integration in the inner integral in (24) one obtains

A�(a, θ, r) =1

4(2πi)n−pn

∫

∂Bn

1∫

0

(θ · ξ)2k+1(1 − t)2k

×w (a, ξ, r(θ · ξ)(1 − t)) sgn(θ · ξ) dt dσξ (25)


Then one introduces a new variable of integration ξ′ instead of ξ in (25) by asuitable orthogonal substitution. Let η be an arbitrary chosen unit vector andconsider θ restricted to the half sphere

∂B+n,η ≡ {θ ∈ ∂Bn : θ · η > 0}. (26)

Let Tη denote the real analytic matrix valued function from ∂B+n,η to Mn(R)

with (j, k) entry Tη,jk defined by

Tη,jk(θ) ≡ δjk + 2θjηk − (θ + η)j(θ + η)k

1 + θ · η∀θ ∈ ∂B

+n,η. (27)

One verifies that Tη(θ) is an orthogonal matrix and that Tη(θ)tθ = η for allθ ∈ ∂B

+n,η. In particular, θ · Tη(θ)ξ′ = η · ξ′ for all ξ′ ∈ R

n and all ∂B+n,η.

Then, by taking ξ = Tη(θ)ξ′ in (25) one obtains

A�(a, θ, r) = 14(2πi)n−pn

∫

∂Bn

1∫

0

w (a, Tη(θ)ξ′, r(η · ξ′)(1 − t))

×(η · ξ′)2k+1(1 − t)2ksgn(η · ξ′) ds dσξ′

for all (a, θ, r) ∈ ER(2k, n)×∂B+n,η ×R. Now, by Lemma 2.1 one deduces that

the restriction of the function A� to ER(2k, n) × ∂B+n,η × R is real analytic.

Since η is an arbitrarily chosen unit vector of Rn, it follows that A� is real

analytic from ER(2k, n) × ∂Bn × R to R.Now consider the function W1. Denote by A and B the functions from

ER(2k, n)×∂Bn ×R to R and from ER(2k, n)×Rn to R, respectively, defined

by

A(a, θ, r) ≡ − 1(2πi)n−pn

∫

∂Bn

w(a, ξ, r(θ · ξ))(θ · ξ)2k log |θ · ξ| dσξ

for all (a, θ, r) ∈ ER(2k, n) × ∂Bn × R, and

B(a, x) ≡ − 1(2πi)n−pn

∫

∂Bn

v(a, x, ξ, 0) dσξ ∀(a, x) ∈ ER(2k, n) × Rn,

where v and w are as in Propositions 3.2 and 3.4. Then, by definition (17),by equality v(a, x, ξ, t) = (x · ξ − t)2kw(a, ξ, x · ξ − t), and by a straightforwardcalculation one verifies that W1(a, x) = |x|2kA(a, x/|x|, |x|)+log |x|B(a, x)for all (a, x) ∈ ER(2k, n)×(Rn\{0}). By Lemma 2.1 one proves that B is realanalytic from ER(2k, n)×R

n to R. Then, by equality ∂αx v(a, 0, ξ, 0) = 0 for all

(a, ξ) ∈ ER(2k, n)×∂Bn and all α ∈ Nn with |α| ≤ 2k−1 (cf. Proposition 3.2),

and by standard theorems on differentiation under integral sign, one showsthat ∂α

x B(a, 0) = 0 for all a ∈ ER(2k, n) and all α ∈ Nn with |α| ≤ 2k − 1.

Now one has to prove that A is real analytic from ER(2k, n) × ∂Bn × R toR. To do so, fix a unit vector η ∈ ∂Bn. Then verify that

A(a, θ, r) = − 1(2πi)n−pn

∫

∂Bn

w(a, Tη(θ)ξ′, r(η · ξ′))(η · ξ′)2k log |η · ξ′| dσξ′


for all (a, θ, r) ∈ ER(2k, n)×∂B+n,η ×R, where Tη(θ) is the orthogonal matrix

introduced in (27). Hence, Lemma 2.1 implies that the restriction of the func-tion A to ER(2k, n) × ∂B

+n,η × R is real analytic. Since η is an arbitrarily

chosen unit vector of Rn, it follows that A is real analytic from ER(2k, n) ×

∂Bn × R to R.Finally consider the function W2. Let v be the function from ER(2k, n)×

Rn × ∂Bn × R to R defined by v(a, x, ξ, t) ≡ (v(a, x, ξ, t) − v(a, x, ξ, 0))/t for

all (a, x, ξ, t) ∈ ER(2k, n) × Rn × ∂Bn × R. Then, by standard properties of

real analytic functions, v is real analytic from ER(2k, n) × Rn × ∂Bn × R to

R. Then set

C(a, x) ≡ − 1(2πi)n−pn

∫

∂Bn

1∫

0

(x · ξ)v(a, x, ξ, (x · ξ)s) ds dσξ

for all (a, x) ∈ ER(2k, n) × Rn. By Lemma 2.1 one deduces that C is real

analytic from ER(2k, n) × Rn to R and a straightforward calculation shows

that W2(a, x) = C(a, x) for all (a, x) ∈ ER(2k, n) × (Rn\{0}). �

5. The Functions S and S0

In the following Theorem 5.1 we introduce a real analytic function S whichsatisfies the conditions in (1) and (2).

Theorem 5.1. Let S be the function from ER(2k, n) × (Rn\{0}) to R definedby

S(a, x) ≡ pnΔ(n+1)/2W0(a, x) + (1 − pn)Δn/2(W1(a, x) + W2(a, x))

for all (a, x) ∈ ER(2k, n)×(Rn\{0}). Then S is real analytic from ER(2k, n)×(Rn\{0}) to R, S(a, ·) is a locally integrable function on R

n for all a ∈ER(2k, n), and S(a, ·) is a fundamental solution of the operator L[a] for alla ∈ ER(2k, n).

Proof. Proposition 4.2 and standard properties of real analytic functions im-ply that S is real analytic. Then, by Proposition 4.2 and by a straightforwardcalculation one verifies that S(a, ·) is a locally integrable function on R

n forall a ∈ ER(2k, n). Hence S(a, ·) defines a distribution on R

n. Finally, byProposition 4.2 and by a standard argument based on the divergence theo-rem one verifies that S(a, ·) = pnΔ(n+1)/2W0(a, ·) + (1 − pn)Δn/2(W1(a, ·) +W2(a, ·)) in the sense of distributions on R

n. Hence Proposition 4.1 impliesthat S(a, ·) = Sa, and thus S(a, ·) is a fundamental solution of L[a] for alla ∈ ER(2k, n). �

In Theorem 5.2 here below we provide a detailed expression for S.

Theorem 5.2. Let S be as in Theorem 5.1. Then, there exist a real analyticfunction A from ER(2k, n) × ∂Bn × R to R, and two real analytic functionsB and C from ER(2k, n) × R

n to R such that

S(a, x) = |x|2k−nA(a, x/|x|, |x|) + log |x|B(a, x) + C(a, x) (28)


for all (a, x) ∈ ER(2k, n) × (Rn\{0}). The functions B and C are identically0 if n is odd and there exist a sequence {fj}j∈N of real analytic functionsfrom ER(2k, n) × ∂Bn to R, and a family {bα}|α|≥sup{k−n,0} of real analyticfunctions from ER(2k, n) to R, such that

fj(a,−θ) = (−1)jfj(a, θ) ∀(a, θ) ∈ ER(2k, n) × ∂Bn,

and

A(a, θ, r) =∞∑

j=0

rjfj(a, θ) ∀ (a, θ, r) ∈ ER(2k, n) × ∂Bn × R, (29)

B(a, x) =∑

|α|≥sup{2k−n,0}bα(a)xα ∀(a, x) ∈ ER(2k, n) × R

n, (30)

where the series in (29) and (30) converge absolutely and uniformly in allcompact subsets of ER(2k, n) × ∂Bn × R and of ER(2k, n) × R

n, respectively.

Proof. The theorem is proved separately for n odd and n even. First assumethat n is odd. By Proposition 4.2, one has W0(a, x) = |x|2k+1A�(a, x/|x|, |x|)with A� real analytic from ER(2k, n) × ∂Bn × R to R. Then, by an inductionargument on the order of differentiation one proves that there exists a realanalytic map A from E (2k, n) × ∂Bn × R to R such that

Δ(n+1)/2W0(a, x) = |x|2k−nA(a, x/|x|, |x|) ∀(a, x) ∈ ER(2k, n) × (Rn\{0})

(see also [10, §4]). Hence, equality (28) for n odd holds with B ≡ 0 and C ≡ 0(cf. Theorem 5.1). To complete the proof for n odd one has to show the exis-tence of the family of functions {fj}j∈N as in the statement. By Proposition3.3, by definition (16), by inequality (15), and by the dominated convergencetheorem, one has W0(a, x) =

∑∞j=2k W0,j(a, x) for all (a, x) ∈ ER(2k, n) ×

(Rn\{0}) with

W0,j(a, x) ≡ − 14(2πi)n−1

∫

∂Bn

aj(a, ξ)(j + 1)!

(x · ξ)j+1 sgn(x · ξ) dσξ

for all (a, x) ∈ ER(2k, n) × (Rn\{0}) and j ∈ N. Let A�j be defined by

A�j(a, θ) ≡ − 1

4(2πi)n−1

∫

∂Bn

aj(a, ξ)(j + 1)!

(θ · ξ)j+1 sgn(θ · ξ) dσξ (31)

for all (a, θ) ∈ ER(2k, n)×∂Bn and j ∈N. Then W0,j(a, x)= |x|j+1A�j(a, x/|x|)

for all (a, x) ∈ ER(2k, n) × (Rn\{0}) and all j ∈ N. We show that A�j is real

analytic. Let η ∈ ∂Bn. Then, one verifies that

A�j(a, θ) = − 1

4(2πi)n−1

∫

∂Bn

aj(a, Tη(θ)ξ′)(j + 1)!

(η · ξ′)j+1 sgn(η · ξ′) dσξ′

for all (a, θ) ∈ ER(2k, n) × ∂B+n,η and all j ∈ N, where Tη(θ) is defined as

in (27) (see also (26)). Hence, by Lemma 2.1 one proves that the restrictionof A�

j to ER(2k, n) × ∂B+n,η is real analytic. Thus A�

j is real analytic fromER(2k, n) × ∂Bn to R for all j ∈ N. Moreover, A�

j(a,−θ) = (−1)jA�j(a, θ)


for all (a, θ) ∈ ER(2k, n) × ∂Bn and all j ∈ N. Then, by inequality (15) andby definition (31) one verifies that the series

∑∞j=0 rjA�

2k+j(a, θ) convergesabsolutely and uniformly in the compact subsets of ER(2k, n) × ∂Bn × R.Hence, by an induction argument on the order of differentiation one provesthat there exist real analytic functions fj from ER(2k, n)×∂Bn to R such that

Δ(n+1)/2(|x|2k+1+jA�

2k+j(a, x/|x|)) = |x|2k−n+jfj(a, x/|x|) (32)

for all (a, x) ∈ ER(2k, n) × (Rn\{0}) and all j ∈ N (see also [10, §4]). Fur-ther, one has fj(a,−θ) = (−1)jfj(a, θ) for all (a, θ) ∈ ER(2k, n) × ∂Bn, j ∈N and the series

∑∞j=0 rjfj(a, θ) converges absolutely and uniformly in the

compact subsets of ER(2k, n) × ∂Bn × R. It follows that Δ(n+1)/2W0(a, x) =|x|2k−n

∑∞j=0 |x|jfj(a, x/|x|) for all (a, x) ∈ ER(2k, n) × (Rn\{0}). Then, by

equality Δ(n+1)/2W0(a, x) = |x|2k−nA(a, x/|x|, |x|) and by standard proper-ties of real analytic functions one has A(a, θ, r) =

∑∞j=0 rjfj(a, θ) for all

(a, θ, r) ∈ ER(2k, n) × ∂Bn × R and the proof for n odd is complete.Now consider the case of dimension n even. By Proposition 4.2, one has

W1(a, x) = |x|2kA(a, x/|x|, |x|) + log |x|B(a, x) with A real analytic fromER(2k, n) × ∂Bn × R to R and B real analytic from ER(2k, n) × R

n to R.Then, by an induction argument on the order of differentiation one verifiesthat there exist real analytic functions A from ER(2k, n)×∂Bn×R to R and Bfrom ER(2k, n)×R

n to R, such that Δn/2W1(a, x) = |x|2k−nA(a, x/|x|, |x|)+log |x|B(a, x) for all (a, x) ∈ ER(2k, n) × (Rn\{0}) (see also [10, §4]). SinceW2(a, x) = C(a, x) for all (a, x) ∈ ER(2k, n) × (Rn\{0}) with C real ana-lytic from ER(2k, n)×R

n to R (cf. Proposition 4.2), one deduces that equality(28) for n even holds with A, B as above and C(a, x) ≡ Δn/2C(a, x) for all(a, x) ∈ ER(2k, n) × R

n (see also Theorem 5.1).To complete the proof in the case of dimension n even, one has to show

the existence of the families of functions {fj}j∈N and {bα}|α|≥sup{2k−n,0} asin the statement. By Proposition 3.3, by definition (17), by the inequalityin (15), and by the dominated convergence theorem, one has W1(a, x) =∑∞

j=2k W1,j(a, x) for all (a, x) ∈ ER(2k, n) × (Rn\{0}), with

W1,j(a, x) ≡ − 1(2πi)n

∫

∂Bn

aj(a, ξ)j!

(x · ξ)j log |x · ξ| dσξ

for all (a, x) ∈ ER(2k, n) × (Rn\{0}) and j ∈ N. Let Aj and B

j be the func-tions defined by

Aj(a, θ) ≡ − 1

(2πi)n

∫

∂Bn

aj(a, ξ)j!

(θ · ξ)j log |θ · ξ| dσξ (33)

for all (a, θ) ∈ ER(2k, n) × ∂Bn and all j ∈ N, and

Bj(a, x) ≡ − 1

(2πi)n

∫

∂Bn

aj(a, ξ)j!

(x · ξ)j dσξ (34)

for all (a, x) ∈ ER(2k, n) × Rn and j ∈ N. Then one has W1,j(a, x) =

|x|jAj(a, x/|x|)+ log |x|B

j(a, x) for all (a, x) ∈ ER(2k, n)× (Rn\{0}) and all


j ∈ N. By arguing so as above for A�j one proves that A

j is real analytic fromER(2k, n) × ∂Bn to R for all j ∈ N. By Lemma 2.1 and Proposition 3.3 oneverifies that also B

j is real analytic from ER(2k, n)× ∂Rn to R for all j ∈ N.

Further, one has Aj(a,−θ) = (−1)jA

j(a, θ) and Bj(a, tx) = tjB

j(a, x) for all(a, x) ∈ ER(2k, n)×R

n, all t ∈ R, and all j ∈ N. Hence one deduces that thereexist real analytic functions b

α from ER(2k, n) to R, for all α ∈ Nn, such that

Bj(a, x) =

∑|α|=j b

α(a)xα for all (a, x) ∈ ER(2k, n) × Rn, j ∈ N. Also note

that bα = 0 if |α| ≤ 2k−1. Then, by the inequality in (15) one proves that the

series∑∞

j=0 rjA2k+j(a, θ, r) and

∑|α|≥2k b

α(a)xα =∑∞

j=0 B2k+j(a, x) con-

verge absolutely and uniformly in the compact subsets of ER(2k, n)×∂Bn ×R

and of ER(2k, n) × Rn, respectively. Now, by an induction argument on the

order of differentiation one verifies that there exist real analytic functions fj

from ER(2k, n) × ∂Bn to R with j ∈ N and real analytic functions bα fromER(2k, n) to R with α ∈ N

n, such that

Δn/2(|x|2k+jA

2k+j(a, x/|x|, |x|) + log |x|B2k+j(a, x)

)

= |x|2k−n+jfj(a, x/|x|) + log |x| ∑

|α|=2k−n+j

bα(a)xα (35)

for all (a, x) ∈ ER(2k, n) × (Rn\{0}) and all j ∈ N (see also [10, §4]). Herewe understand that

∑|α|=2k−n+j bα(a)xα = 0 if 2k − n + j < 0. Further,

one has fj(a,−θ) = (−1)jfj(a, θ) for all (a, θ) ∈ ER(2k, n) × ∂Bn, andall j ∈ N, and bα = 0 if |α| < 2k − n. The series

∑∞j=0 rjfj(a, θ) and

∑|α|≥sup{2k−n,0} bα(a)xα converge absolutely and uniformly in the compact

subsets of ER(2k, n)×∂Bn ×R and of ER(2k, n)×Rn, respectively. Thus one

verifies that

Δn/2W1(a, x) = |x|2k−n∞∑

j=0

|x|jfj(a, x/|x|) + log |x|∑

|α|≥sup{2k−n,0}bα(a)xα

for all (a, x) ∈ ER(2k, n) × (Rn\{0}). Then, by equality Δn/2W1(a, x) =|x|2k−nA(a, x/|x|, |x|)+ log |x|B(a, x) and by standard properties of real ana-lytic functions one verifies that A(a, θ, r) =

∑∞j=0 rjfj(a, θ) for all (a, θ, r) ∈

ER(2k, n) × ∂Bn × R and that B(a, x) =∑

|α|≥sup{2k−n,0} bα(a)xα for all(a, x) ∈ ER(2k, n) × R

n. The proof of the theorem is now complete. �

In the following Theorem 5.3 we introduce a real analytic function S0

from ER(2k, n) × (Rn\{0}) to R such that S0(a, ·) is a fundamental solutionof the principal term L0[a] of the operator L[a], for all a ∈ ER(2k, n) (seealso (4)).

Theorem 5.3. Let {fj}j∈N and {bα}|α|≥sup{2k−n,0} be as in Theorem 5.2. LetS0 be the function from ER(2k, n) × (Rn\{0}) to R defined by

S0(a, x) ≡ |x|2k−nf0(a, x/|x|) + log |x|∑

|α|=2k−n

bα(a)xα


for all (a, x) ∈ ER(2k, n) × (Rn\{0}). Then S0(a, ·) is a fundamental solu-tion of the homogeneous operator L0[a] for all fixed a ∈ ER(2k, n) (note that∑

|α|=2k−n bα(a)xα = 0 if n is odd or ≥ 2k + 1).

Proof. Let v0 be the function from ER(2k, n)× Rn × ∂Bn × R to R defined by

v0(a, x, ξ, t) ≡ a2k(a, ξ)(2k)!

(x · ξ − t)2k =(x · ξ − t)2k

(2k)!P0[a](ξ)(36)

for all (a, x, ξ, t) ∈ ER(2k, n)×Rn ×∂Bn ×R (cf. Proposition 3.3). Then one

verifies that v0 is the unique real analytic function from ER(2k, n) × Rn ×

∂Bn × R to R such that L0[a]v0(a, x, ξ, t) = 1 for all (a, x, ξ, t) ∈ ER(2k, n) ×R

n × ∂Bn × R and that ∂αx v0(a, x, ξ, t) = 0 for all (a, x, ξ, t) ∈ ER(2k, n) ×

Rn × ∂Bn × R with x · ξ = t and all α ∈ N

n with |α| ≤ 2k − 1 (see alsoProposition 3.2). Then one introduces the functions W0, W1, and W2 fromER(2k, n) × (Rn\{0}) to R defined by

W0(a, x) ≡ 14(2πi)n−pn

∫

∂Bn

x·ξ∫

0

v0(a, x, ξ, t) sgn t dt dσξ, (37)

W1(a, x) ≡ − 1(2πi)n−pn

∫

∂Bn

v0(a, x, ξ, 0) log |x · ξ| dσξ, (38)

W2(a, x) ≡ − 1(2πi)n−pn

∫

∂Bn

x·ξ∫

0

v0(a, x, ξ, t) − v0(a, x, ξ, 0)t

dt dσξ (39)

for all (a, x) ∈ ER(2k, n) × (Rn\{0}). If the dimension n is odd, then The-orem 5.1 implies that the function Δ(n+1)/2W0(a, ·) is a fundamental solu-tion of L0[a], for all a ∈ ER(2k, n). Moreover, by (31), (32), (36), (37),and by a straightforward calculation, one verifies that Δ(n+1)/2W0(a, x) =|x|2k−nf0(a, x/|x|) for all (a, x) ∈ ER(2k, n) × (Rn\{0}). Hence the validityof the theorem for n odd is proved. Now let n be even. Then one can verifythat W2(a, ·) equals a polynomial function of degree 2k on R

n, for all fixeda ∈ ER(2k, n). Thus L0[a]Δn/2W2(a, ·) = 0 because L0[a]Δn/2 is an homoge-neous operator of order 2k+n. Hence, Theorem 5.1 implies that the functionΔn/2W1(a, ·) is a fundamental solution of L0[a] for all a ∈ ER(2k, n). More-over, by (33), (34), (35), (36), (38), and by a straightforward calculation, oneverifies that Δn/2W1(a, x) = |x|2k−nf0(a, x/|x|) + log |x|∑|α|=2k−n bα(a)xα

for all (a, x) ∈ ER(2k, n) × (Rn\{0}). Now the proof is complete. �

6. The Single Layer Potential v[a, µ]

In this section we show some properties of the single layer potential v[a, μ]corresponding to the fundamental solution S(a, ·) introduced in Theorem5.2 [see also definition (5)]. We observe that only the expression for S(a, x)obtained in Theorem 5.2 is here exploited and not the joint real analyticdependence of S(a, x) upon a ∈ ER(2k, n) and upon x ∈ R

n\{0}. However,the results which we show in this section for v[a, μ] together with the joint


real analyticity of S play an important role in the proof obtained in [9] for thejoint real analytic dependence of v[a, μ] upon a, upon μ, and upon a suitablefunction φ which parameterizes the support of integration ∂Ω (see also Sec-tion 1 here above and references therein). In particular, the simple separatereal analytic dependence of S(a, x) upon a ∈ ER(2k, n) and x ∈ R

n\{0} isnot sufficient in the argument of [9]. We also observe that for an operator ina Banach space, even in the finite dimensional case, separate real analyticityon all the variables does not imply real analyticity (cf. Boman [1]).

We use the following notation. Let a ∈ ER(2k, n). Let μ ∈ Cm−1,λ(∂Ω).Let β ∈ N

n and |β| ≤ 2k − 1. Then vβ [a, μ] denotes the function from Rn to

R defined by

vβ [a, μ](x) ≡∫

∂Ω

∂βx S(a, x − y)μ(y) dσy ∀x ∈ R

n,

where the integral is understood in the sense of singular integrals if x ∈ ∂Ωand |β| = 2k − 1. Thus, for β = (0, . . . , 0) one has v(0,...,0)[a, μ] = v[a, μ].Moreover, by standard theorems on differentiation under integral sign oneverifies that

vβ [a, μ](x) = ∂βx v[a, μ](x) ∀x ∈ R

n\∂Ω, β ∈ Nn, |β| ≤ 2k − 1. (40)

The following Theorem 6.1 implies that v[a, μ] belongs to C2k−2(Rn) andthat ∂β

xv[a, μ] = vβ [a, μ] in the whole of Rn for all β ∈ N

n with |β| ≤ 2k − 2.

Theorem 6.1. Let a ∈ ER(2k, n). Let μ ∈ Cm−1,λ(∂Ω). Let β ∈ Nn and

|β| ≤ 2k − 2. Then, vβ [a, μ] ∈ C2k−2−|β|(Rn) and ∂βxv[a, μ](x) = vβ [a, μ](x)

for all x ∈ Rn.

Proof. By Theorem 5.2, by standard properties of real analytic functions,and by an induction argument on the order of differentiation one verifiesthat for all α ∈ N

n there exist real analytic functions Aα and Bα fromER(2k, n)×∂Bn ×R to R such that ∂α

x S(a, x) = |x|2k−n−|α|Aα(a, x/|x|, |x|)+|x|2k−n−|α| log |x|Bα(a, x/|x|, |x|) + ∂α

x C(a, x) for all (a, x) ∈ ER(2k, n) ×(Rn\{0}) (see also [10, §4]). In particular, ∂α

x S(a, x) = o(|x|2−n−1/2) as|x| → 0+ for all α ∈ N

n with |α| ≤ 2k − 2. Then, by the Vitali ConvergenceTheorem one deduces that the function vα[a, μ] is continuous on R

n for allα ∈ N

n, |α| ≤ 2k−2 (see also Folland [12, Proposition 3.25] where the conti-nuity of the single layer potential corresponding to the fundamental solutionof the Laplace operator is proved, but the proof for vα[a, μ] is based on thesame argument). Also, a standard argument based on the divergence theoremshows that ∂α

x v[a, μ] = vα[a, μ] in the sense of distributions on Rn. Then, by

a standard argument based on the convolution with a family of mollifiers itfollows that v[a, μ] ∈ C2k−2(Rn) and that ∂α

x v[a, μ](x) = vα[a, μ](x) for allx ∈ R

n and for all α ∈ Nn with |α| ≤ 2k−2. Thus ∂β

xv[a, μ] ∈ C2k−2−|β|(Rn)and the validity of the theorem is proved. �

As we have seen in Theorem 6.1, v[a, μ] is a function of class C2k−2.In Theorem 6.6 here below we show that the restriction of v[a, μ] to Ω canbe extended to a function of class Cm+2k−2,λ from clΩ to R (a similar result


holds for the restriction of v[a, μ] to the exterior domain Rn\clΩ in a local

sense which will be clarified). In order to prove Theorem 6.6 we exploit anidea of Miranda (cf. [23, §5]). Accordingly, we first state in Theorem 6.2 aresult by Miranda (cf. [23, Theorem 2.1]).

Theorem 6.2. Let K ∈ C2m(Rn\{0}) be such that

K(x) = |x|1−nK(x/|x|) ∀x ∈ Rn\{0}, (41)

and∫

Π∩∂Bn

K(η) dση = 0 for every hyper-plane Π of Rn which contains 0. (42)

Let p[μ] be the function from Rn\∂Ω to R defined by

p[μ](x) ≡∫

∂Ω

K(x − y)μ(y) dσy ∀x ∈ Rn\∂Ω

for all μ ∈ Cm−1,λ(∂Ω). Then the following statements hold.(i) The restriction p[μ]|Ω extends to a unique continuous function p+[μ]

on clΩ for all μ ∈ Cm−1,λ(∂Ω). The map which takes μ to p+[μ] iscontinuous from Cm−1,λ(∂Ω) to Cm−1,λ(clΩ).

(ii) The restriction p[μ]|Rn\clΩ extends to a unique continuous function p−[μ]on R

n\Ω for all μ ∈ Cm−1,λ(∂Ω). If R ∈]0,+∞[ and clΩ is contained inRBn, then the map which takes μ to p−[μ]|cl(RBn)\Ω is continuous fromCm−1,λ(∂Ω) to Cm−1,λ(cl(RBn)\Ω).

We now introduce in the following Lemmas 6.3–6.5 some technical re-sults which are needed in the proof of Theorem 6.6. The proofs of Lemmas6.3–6.5 can be deduced by elementary induction arguments and are accord-ingly omitted.

Lemma 6.3. Let j, q ∈ N, q ≥ 1. Let α ∈ Nn and |α| = q − 1. Let f be a

real analytic function from ∂Bn to R such that f(−θ) = (−1)jf(θ) for allθ ∈ ∂Bn. Let K(x) ≡ ∂α

x (|x|q−nf(x/|x|)) for all x ∈ Rn\{0}. If the sum j + q

is even, then K satisfies the conditions in (41) and (42).

Lemma 6.4. Let j ∈ N. Let α ∈ Nn and |α| = n + j − 1. Let p be a real

homogeneous polynomial function from Rn to R of degree j. Let K(x) ≡

∂αx (p(x) log |x|) for all x ∈ R

n\{0}. If the dimension n is even, then K sat-isfies the conditions in (41) and (42).

Lemma 6.5. Let j ∈ N, j ≥ 1. Let F , G be real analytic functions from∂Bn × R to R. Let H be the function from R

n\{0} to R defined by H(x) ≡|x|jF (x/|x|, |x|)+ |x|j log |x|G(x/|x|, |x|) for all x ∈ R

n\{0}. Then H extendsto an element of Cj−1(Rn).

Then we have the following Theorem 6.6.

Theorem 6.6. Let a ∈ ER(2k, n). Then the following statements hold.(i) The map which takes μ to v[a, μ]|clΩ is continuous from Cm−1,λ(∂Ω) to

Cm+2k−2,λ(clΩ).


(ii) If R > 0 and clΩ ⊆ RBn, then the map which takes μ to v[a, μ]|cl(RBn)\Ω

is continuous from Cm−1,λ(∂Ω) to Cm+2k−2,λ(cl(RBn)\Ω).

Proof. Let A, B, C, {fj}j∈N, {bα}|α|≥sup{2k−n,0} be as in Theorem 5.2. Then,by standard properties of real analytic functions one has

S(a, x) =m+n−1∑

j=0

fj(a, x/|x|)|x|2k−n+j +∑

2k−n≤|α|≤m+2k−1

bα(a)xα log |x|

+ |x|m+2kAa,m(x/|x|, |x|) + |x|m+2k log |x|Ba,m(x/|x|, |x|) + C(a, x) (43)

for all x ∈ Rn\{0}, where Aa,m and Ba,m are real analytic functions from

∂Bn × R to R.Consider the terms fj(a, x/|x|)|x|2k−n+j which appear in the left hand

side of (43). By Lemma 6.3, the function which takes x ∈ Rn\{0} to

∂βx (fj(a, x/|x|)|x|2k−n+j) satisfies the conditions in (41) and (42) for all

β ∈ Nn with |β| = 2k − 1 + j. Hence, by Theorem 6.2, by the continuous

embedding of Cm+2k−2+j,λ(clΩ) in Cm+2k−2,λ(clΩ), and by the classical the-orem on differentiation under integral sign one deduces that the map fromCm−1,λ(∂Ω) to Cm+2k−2,λ(clΩ) which takes μ to the unique extension to clΩof

∫

∂Ω

fj (a, (x − y)/|x − y|) |x − y|2k−n+jμ(y) dσy ∀x ∈ Ω

is continuous. Similar result one has if clΩ is replaced by cl(RBn)\Ω.Now consider the terms bα(a)xα log |x| which appear in the left hand

side of (43). By Lemma 6.4 one verifies that ∂βx (bα(a)xα log |x|) satisfies the

conditions in (41) and (42) for all β ∈ Nn with |β| = n + |α| − 1. Then,

by Theorem 6.2, by the continuous embedding of Cm−1+(n+|α|−1),λ(clΩ) inCm+2k−2,λ(clΩ), and by the classical theorem on differentiation under inte-gral sign one proves that the map from Cm−1,λ(∂Ω) to Cm+2k−2,λ(clΩ) whichtakes μ to the unique extension to clΩ of

∫

∂Ω

bα(a) (x − y)α log |x − y|μ(y)dσy ∀x ∈ Ω

is continuous. Similar result one has if clΩ is replaced by cl(RBn)\Ω.Finally, note that the term

|x|m+2kAa,m(x/|x|, |x|) + |x|m+2k log |x|Ba,m(x/|x|, |x|)in the left hand side (43) is a function of the form considered in Lemma 6.5and hence extends to an element of Cm+2k−1(Rn). Moreover C(a, ·) is realanalytic from R

n to R and thus in Cm+2k−1(Rn).Hence, by equality (43), and by standard theorems on differentiation

under integral sign, and by the continuity of the embedding of Cm+2k−1(Rn)into Cm+2k−2,λ(Rn), one completes the proof of the theorem. �

As an immediate consequence of Theorem 6.6 and of equality (40), oneverifies the validity of the following Corollary 6.7.


Corollary 6.7. Let a ∈ ER(2k, n). Let μ ∈ Cm−1,λ(∂Ω). Let β ∈ Nn, |β| =

2k − 1. Then the following statements hold.(i) The restriction vβ [a, μ]|Ω extends to a continuous function v+

β [a, μ] onclΩ which belongs to Cm−1,λ(clΩ).

(ii) The restriction vβ [a, μ]|Rn\clΩ extends to a continuous function v−β [a, μ]

on Rn\Ω and v−

β [a, μ]|cl(RBn)\Ω ∈ Cm−1,λ(cl(RBn)\Ω) for all R > 0such that clΩ ⊆ RBn.

Finally, we describe in the following Theorem 6.8 the jump propertiesof the single layer potential v[a, μ].

Theorem 6.8. Let a ∈ E (2k, n). Let μ ∈ Cm−1,λ(∂Ω). Let β ∈ Nn and

|β| = 2k − 1. Let v+β [a, μ] and v−

β [a, μ] be as in Corollary 6.7. Then

v±β [a, μ](x) = ∓ νΩ(x)β

2P0[a](νΩ(x))μ(x) + vβ [a, μ](x) ∀x ∈ ∂Ω, (44)

where νΩ denotes the outward unit normal to the boundary of Ω.

Proof. Let S0(a, ·) be the function in Theorem 5.2 and

v0,β [a, μ](x) ≡∫

∂Ω

∂βxS0(a, x − y)μ(y) dσy ∀x ∈ R

n\∂Ω.

Since S0(a, ·) is a fundamental solution of the homogeneous operator L0[a],Corollary 6.7 implies that v0,β [a, μ]|Ω and v0,β [a, μ]|Rn\clΩ extend to uniquecontinuous functions v+

0,β [a, μ] on clΩ and v−0,β [a, μ] on R

n\Ω, respectively.Moreover,

v±0,β [a, μ](x) = ∓ νΩ(x)β

2P0[a](νΩ(x))μ(x) + v0,β [a, μ](x) ∀x ∈ ∂Ω (45)

(cf., e.g., Mitrea [22, pp. 392–393], see also Cialdea [3, §2, IX], [4, Theo-rem 3]). Now let

S∞(a, x) ≡ S(a, x) − S0(a, x) ∀x ∈ Rn\{0}

and define

v∞,β [a, μ](x) ≡∫

∂Ω

∂βxS∞(a, x − y)μ(y) dσy ∀x ∈ R

n\∂Ω.

So that

vβ [a, μ] = v0,β [a, μ] + v∞,β [a, μ]. (46)

By Theorem 5.2 and by a standard properties of real analytic functions, onehas S∞(a, x) = |x|2k+1−nA∞(a, x/|x|, |x|)+ |x|2k+1−n log |x|B∞(x/|x|, |x|)+C(a, x) for all x ∈ R

n\{0}, where A∞ and B∞ are real analytic functionsfrom ER(2k, n)×∂Bn ×R to R. Then, by an induction argument on the orderof differentiation one verifies that ∂β

xS∞(a, x) = |x|2−nA∞,β(a, x/|x|, |x|) +|x|2−n log |x|B∞,β(a, x/|x|, |x|) + ∂β

x C(a, x) for all x ∈ Rn\{0}, where A∞,β

and B∞,β are real analytic functions from ER(2k, n)×∂Bn ×R to R (see also


[10, §4]). Then, by the Vitali Convergence Theorem one proves that the func-tion v∞,β [a, μ] is continuous on R

n (see also Folland [12, Proposition 3.25]where the continuity of the single layer potential corresponding to the funda-mental solution of the Laplace operator is proved, but the proof for v∞,β [a, μ]is based on the same argument). Hence, by the equalities in (45) and (46)one deduces the validity of (44). �

Acknowledgments

The author wishes to thank the anonymous referee for the valuable commentswhich have improved the paper by showing that the existence of a functionS which satisfies the conditions in (1), (2) can be deduced by the results ofMantlik in [20,21].

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Matteo Dalla RivaCentro de Investigacao e Desenvolvimento em Matematica e Aplicacoes (CIDMA)Universidade de Aveiro3810-193 AveiroPortugale-mail: [email protected]

Received: June 20, 2012.

Revised: March 19, 2013.

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