Norbert Ortner, Peter Wagner€¦ · ducts of barrelled (DF)-spaces, a category of locally...

Norbert Ortner, Peter Wagner

Distribution-Valued Analytic FunctionsTheory and Applications

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Distribution-Valued Analytic FunctionsTheory and Applications

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Distribution-Valued AnalyticFunctions

Theory and Applications

edition swk

Dr. Norbert Ortner, Dr. Peter WagnerInstitut für MathematikUniversität InnsbruckTechnikerstr. 13, A-6020 Innsbruckemail: [email protected]

c© N. Ortner, P. Wagner, 2013

Erschienen in der edition swk (www.stiftung-swk.de/edition-swk)Co-Verlag: tredition GmbH, Grindelallee 188, 20144 HamburgPrinted in GermanyISBN: 978-3-8491-1968-3

Mathematics Subject Classification (2010): 46Fxx, 46F10

Detailed bibliographical information on this publication can be found at http://dnb.d-nb.de.

All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (tredition GmbH, Grindelallee 188, 20144 Hamburg),except for brief excerpts in connection with reviews or scholarly analysis. Use in connectionwith any form of information storage and retrieval, electronic adaption, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.

Cover design: Tamara Pulkert, Nürnberg, Germany.(The cover graphic shows a section of the function 1/|Γ | on the left half-plane, represented with differently

scaled axes.)

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Distributions on Hypersurfaces and Pullbacks of Distributions . . . . . . . . . . . 21.2 Convolution of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Convolution of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Continuity of Bilinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Vector-Valued Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.6 Distribution-Valued Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Quasihomogeneous Distributions and Their Fourier Transforms . . . . . . . . . . 312.1 Definition of Quasihomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Representation in Generalized Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 352.3 Powers of Polynomials, Solution of Division Problems . . . . . . . . . . . . . . . . . 402.4 Fundamental Solutions of Iterated Wave Operators . . . . . . . . . . . . . . . . . . . . . 462.5 The Structure Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.6 The Fourier Transform of Quasihomogeneous Distributions . . . . . . . . . . . . . 622.7 Fundamental Solutions and Convolution Kernels . . . . . . . . . . . . . . . . . . . . . . 66

3 Convolution With Quasihomogeneous Distributions . . . . . . . . . . . . . . . . . . . . . . 853.1 Weighted D′

Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.2 Convolution With Homogeneous Distributions With Lp-Characteristics . . . 903.3 Convolution With Homogeneous Distributions With Non-Vanishing

C∞-Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.4 Characterization of the Convolvability of Two Homogeneous Distributions

With C∞-Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 The Convolution Group of Hyperbolic Operators . . . . . . . . . . . . . . . . . . . . . . 1043.6 Convolution for Characteristics in Spaces of Bessel Potentials . . . . . . . . . . . 1083.7 The Convolution Group of Quasihyperbolic Operators . . . . . . . . . . . . . . . . . . 1143.8 Convolution Groups of Singular Integral Operators . . . . . . . . . . . . . . . . . . . . 117

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Introduction

Let us illustrate first, by a simple example (namely the construction of a fundamentalsolution of the Laplace operator ∆n in Rn), the occurence of analytic distribution-valuedfunctions.

Analytic continuation of the differentiation formula

∆n(|x|λ+2) = (λ+ 2)(λ+ n)|x|λ, (0.1)

which holds classically for Reλ > 0, yields, for n 6= 2, the fundamental solution

E =1

(2− n)|Sn−1| |x|2−n

of ∆n. In fact, when we extend the holomorphic distribution-valued function

λ ∈ C; Reλ > −n −→ D′(Rn) : λ 7−→ |x|λ ∈ L1loc(R

n)

by means of equation (0.1) analytically to the left of the line Reλ = −n, then poles appearin λ = −n − 2k, k ∈ N0, and ∆n|x|2−n = (2 − n)|Sn−1| δ results from the Laurentexpansion of |x|λ around −n, i.e.,

|x|λ =|Sn−1|λ+ n

δ + Tλ, Tλ holomorphic for |λ+ n| < 2,

in the following way:

∆n|x|2−n = limλ→−n

∆n|x|λ+2 = limλ→−n

(λ+ 2)(λ+ n)|x|λ

= limλ→−n

(λ+ 2)[|Sn−1| δ + (λ+ n)Tλ

]

= (2− n)|Sn−1| δ.

(Essentially, the above deduction goes back to Riesz [1], [2], and was formulated withdistributions in Dieudonné [3], Ex. (17.9.2), pp. 262-265.)

The aim of this book consists in giving a systematic and general approach to treatingmeromorphic distribution-valued functions of the form λ 7−→ Fλ · λ, where the “charac-teristic" Fλ ∈ D′(Γ ) (with Γ = x ∈ Rn; (x) = 1) also depends meromorphically onλ. (In the example above, we have (x) = |x| and Fλ = 1.)

Let us describe now the contents of the book more in detail. Chapter 1 consists ofsupplements to the theories of locally convex topological vector spaces and, in particular, ofdistribution spaces. Hereby, results from the books Schwartz [5], Robertson and Robertson

viii Introduction

[1], Horváth [4] and Treves [1] are taken for granted and are quoted only. We supplementthese basic references by synopses on distributions on hypersurfaces (1.1), on convolution ofmeasures and distributions (1.2, 1.3), on bilinear mappings defined on barrelled DF−spaces(1.4), and on holomorphic functions with values in topological vector spaces (1.5, 1.6).

In Chapter 2, the quasihomogeneous distribution-valued functions λ 7−→ Fλ · λ aredefined and their properties (analytic continuation, poles, residues, finite parts) are derived(2.1, 2.2). The structure of quasihomogeneous distributions and of their Fourier transformsis elucidated in 2.5, 2.6. In the remaining sections of Chapter 2, the theory is illustrated byseveral concrete examples originating from the quasihomogeneous polynomials

xm11 + · · ·+ xmn

n , x1 + ix2,

ix1 +R(x′), x21 − |x′|2, |x|2, |x′|2 − |x′′|2.Fundamental solutions of the iterated Cauchy-Riemann operator (∂1 + i∂2)

l, of the iteratedwave operator (∂21 −∆n−1)

l, and, more generally, of the iterated ultrahyperbolic operator

(∂21 + · · ·+ ∂2m − ∂2m+1 − · · · − ∂2n)l

are deduced therefrom (Ex. 2.7.3, Prop. 2.4.2, Prop. 2.7.6).In Chapter 3, the convolution with the quasihomogeneous distributions arising in Chapter

2 is treated. For this purpose, we define weighted D′Lp-spaces, which generalize the spaces

D′Lp introduced by L. Schwartz (3.1). The homogeneous distributions F · λ operate on

weighted D′Lp−spaces by convolution, and we obtain continuity properties in dependence

on the regularity of the characteristic F (see 3.2 for F ∈ Lp(Sn−1), 3.3, 3.4 for F ∈C∞(Sn−1), 3.6 for F ∈ Lps(S

n−1)). As application, we describe the convolution groupsof elliptic (3.3), hyperbolic (3.5), ultrahyperbolic (3.6) and quasihyperbolic operators (3.7).Finally, the convolution groups of some singular integral operators are treated in 3.8.

We wish to take this opportunity to express our deep gratitude to Prof. John Horváth,who encouraged and accompanied the writing of this book from the beginning. We aremost grateful to Prof. E. Zeidler for having included an earlier version of this book in thelecture notes series of the MPI Leipzig. We are deeply indebted to Prof. R. Brigola for hiswillingness to publish this book in the edition swk, as well as for his generous help in theprocess of editing.

Innsbruck, January 2013 Norbert Ortner, Peter Wagner

Basic Notation

As usual, N = 1, 2, 3, . . . , N0 = 0 ∪ N, and Z,R,C denote the sets of integer, real,and complex numbers, respectively; R+ = (0,∞) ⊂ R, Sn−1 = x ∈ Rn; |x| = 1 is the(n− 1)-dimensional unit sphere and |Sn−1| = 2πn/2/Γ (n/2) its surface.

For polynomials and differential operators, multiindex notation is used: xα = xα11 . . . xαn

n

for α ∈ Nn0 , ∂α = ∂α1

1 . . . ∂αnn , ∂j = ∂/∂xj , ∆n = ∂21 + · · · + ∂2n, |α| = α1 + · · · + αn,

α! = α1! . . . αn!. The Heaviside function is abbreviated by Y, i.e. Y (x) = 1 for x > 0 andY (x) = 0 else.

For Ω ⊂ Rn open, we use the classical function spaces

E(Ω) = C∞(Ω) = f : Ω −→ C; f is C∞,

Lp(Ω) = f : Ω −→ C measurable; ‖f‖pp =∫

Ω

|f(x)|p dx <∞/ ∼

(where 1 ≤ p <∞ and f ∼ g if f = g almost everywhere),

L∞(Ω) = f : Ω −→ C measurable; ‖f‖∞ = supx∈Ω

|f(x)| <∞/ ∼

L1loc(Ω) = f : Ω −→ C measurable; ∀K ⊂ Ω compact :

∫

K

|f(x)| dx <∞/ ∼ and

L1loc,+(Ω) = f : Ω −→ [0,∞) measurable; ∀K ⊂ Ω compact :

∫

K

|f(x)| dx <∞/ ∼ .

Furthermore, the spaces of test functions D(Ω),D := D(Rn),S,DLp ,.B,OC along with

their corresponding dual distribution spaces D′(Ω),D′,S ′,D′Lq ,D′

L1 ,O′C are as explained

in Schwartz [5]. The angle brackets 〈 , 〉 are used for the dual pairing in locally convextopological vector spaces and, in particular, in distribution spaces. The Fourier transform Fis defined on integrable functions by

(Ff)(x) =∫

Rn

f(ξ)e−iξ·x dξ, ξ · x = ξ1x1 + · · ·+ ξnxn,

and extended to F : S ′ −→ S ′ by continuity.

Chapter 1

Preliminaries

The decomposition of a homogeneous distribution T ∈ D′(Rn) into a “radial" factor λ anda “characteristic" F ∈ D′(Γ ) (where Γ = −1(1)) plays a crucial rôle. Hence, in Section1.1, the spaces D(Γ ), Lp(Γ ), 1 ≤ p ≤ ∞, and D′(Γ ) are defined. Furthermore, we repeatthe definition of the pullback f∗T ∈ D′(Γ ) of a distribution T ∈ D′(R) under a mappingf : Γ −→ R.

In Section 1.2, we introduce the most general definition of convolution of measures

on Rn (according to N. Bourbaki): Two measures µ, ν ∈ M(Rn) are convolvable iffϕ∆(µ ⊗ ν) is an integrable (classically often called “bounded") measure on R2n for eachϕ ∈ K(Rn). The space M1(Rn) of integrable measures, which can be defined as the dualspace of C0(Rn), is shown to be also the dual space of BCb(Rn), the space of boundedcontinuous functions endowed with the so-called Buck topology. This topology can bedescribed explicitly by the seminorms

BC −→ R : f 7−→ ‖f · g‖∞, g ∈ C0(Rn).

The proof of (BCb)′ = M1 (see Prop. 1.2.1) is new as is the characterization of convolvabi-lity in Prop. 1.2.4. If τ(E,F ) denotes the Mackey topology on E with respect to the dualsystem (E,F ), then the Buck topology is just τ(BC,M1).

The definition of convolvability of measures and the different characterizations of it arepresented as an introductory model of the convolvability of distributions in Section 1.3: Twodistributions S, T ∈ D′(Rn) are called convolvable if ϕ∆(S ⊗ T ) is an integrable distribu-tion for each test function ϕ ∈ D(Rn) (L. Schwartz, 1954). The proof of the equivalenceof the various characterizations of convolvability in Prop. 1.3.4 is new as well as condition(v), the generalization in Remark 4 to Prop. 1.3.4, and Lemma 1.3.5.

In Section 1.4, we arrange some results on the continuity of bilinear mappings on pro-ducts of barrelled (DF)-spaces, a category of locally topological vector spaces introducedby A. Grothendieck. The proofs (of Prop. 1.4.1, Prop. 1.4.3, and Lemma 1.4.4) are formu-lated in the language of seminorms.

In Section 1.5, we repeat and complement the theory of “vector-valued" holomorphicfunctions (i.e. holomorphic functions with values in locally convex vector spaces) developedoriginally in three beautiful papers:

L. Schwartz: Espaces de fonctions différentielles à valeurs vectorielles;A. Grothendieck: Sur certains espaces de fonctions holomorphes;J. Horváth: Distribuciones definidas por prolongación analítica.

Besides the standard assertions on the equivalence of strong and of weak holomorphy(Prop. 1.5.2), the analytic continuation (Prop. 1.5.3) and the representation of analyticfunctions by Taylor series (Prop. 1.5.4), we emphasize in Prop. 1.5.6 the analyticity of

2 1 Preliminaries

bilinear mappings u applied to holomorphic functions, i.e. of z 7→ u(f(z), g(z)), under therelatively weak assumption of separate continuity of u, in contrast to that of continuity(cf. e.g. H. G. Garnir, M. de Wilde, J. Schmets). Whereas holomorphic functions f :U −→ D′(Ω) with range in E ′(Ω) (i.e. f(U) ⊂ E ′(Ω)) are indeed holomorphic functionsinto E ′(Ω) (Prop. 1.5.9), this is not necessarily true if E ′ is replaced by other distributionspaces as e.g. S ′. The counterexample in Ex. 1.5.10 is new. The assertions and formulasfor finite parts Pf

z=z0u(f(z), g(z)) and residues Res

z=z0u(f(z), g(z)) for separately continuous

mappings u and meromorphic functions f, g are new and generalize the correspondingresults in the special case of u being scalar multiplication.

In Section 1.6, we specialize the results of Section 1.5 from general locally convexspaces to distribution spaces. In Prop. 1.6.2, the known formulae for ∂α Res

z=z0f(z) and

F(Resz=z0

f(z))

are complemented by a formula for g∗(Resz=z0

f(z)). Props. 1.6.3 and 1.6.4

are new: The assertion of Prop. 1.5.6 is applied to the bilinear mappings which representthe multiplication of distributions with test functions and the convolution of distributions (invarious fixed spaces), respectively. Subsequently, we investigate the question of holomorphicdependence of the convolution of holomorphic distribution-valued functions. The holomor-phy of z 7−→ f(z)∗g(z) is not necessarily implied by the convolvability of f(z) and g(z) foreach z and the holomorphy of the factors f, g : In Prop. 1.6.5 we prove that z 7−→ f(z)∗g(z)is holomorphic if an additional continuity property is satisfied, and in Ex. 1.6.6 we give anexample of two distributions depending holomorphically on z, the convolution of which isnot holomorphic in z.

1.1 Distributions on Hypersurfaces and Pullbacks of Distributions

In contrast to the general theory of distributions on manifolds, i.e. currents, developed by deRham [2]; Schwartz [5], Ch. IX (“Courants sur une variété"); Dieudonné [3], Ch. XVII(“Calcul différentiel sur une variété différentielle"), we shall restrict ourselves to thedefinition of these notions in the very particular cases which are used in Chs. 2 and 3.

Let a1 > 0, . . . , an > 0, and : Rn \ 0 −→ R+ be infinitely differentiable andquasihomogeneous of degree 1 with respect to (a1, . . . , an), i.e.,

∀ t > 0 : ∀ x ∈ Rn : (ta x) = t(x) ,

where ta := (ta11 , . . . , tann ) and x y := (x1y1, . . . , xnyn) for x, y ∈ Rn. Then Γ :=

−1(1) = x ∈ Rn; (x) = 1 is a C∞ submanifold of Rn, which we orient by thepullback to Γ of the Kronecker - Leray form (cf. Krée [1], p. 7)

σ =

n∑

j=1

(−1)j−1ajxjdx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxn ∈ Ωn−1(Rn), (1.1)

wherein Ωk(M) denotes, as usual, the vector space of C∞ k-forms on a C∞ manifold M.Let us remark that

1.1 Distributions on Hypersurfaces and Pullbacks of Distributions 3

d ∧ σ = dx1 ∧ · · · ∧ dxn = dx ∈ Ωn(Rn \ 0)

by Euler’s equation (see (2.1.2)), and hence ι∗σ does not vanish on Γ if ι : Γ → Rn denotesthe canonical injection. Moreover, for Ω ∈ Ωn−1(Rn), we have ι∗Ω = ι∗σ ∈ Ωn−1(Γ )iff d ∧Ω = dx1 ∧ · · · ∧ dxn holds on Γ, i.e. ι∗Ω is determined uniquely by this equation.E.g. this holds if

Ω =

n∑

j=1

(−1)j−1

∂∂xj

|∇|2 dx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxn ∈ Ωn−1(Rn \ 0),

cf. Dautray and Lions [2], pp. 485, 486; Fedoryuk [1], (1.31), p. 105.Integrating (n − 1)-forms over Γ we shall always use the orientation induced by σ, or,

put differently, we shall identify in this wayΩn−1(Γ ) with the space of measures on Γ withC∞ densities with respect to σ.

The equation ι∗σ =

n∑

i=1

aixiνido connecting σ with the surface measure do and the

outward unit normal vector ν =∇|∇| is at the origin of the equivalence of Gauß’ and

Stokes’ theorems: If ϕ ∈ D(Rn), then d(ϕ ·σ) = div(ϕ ·u)dx1 ∧ · · · ∧ dxn, where u(x) =

a x, and therefore application of Gauß’ and of Stokes’ theorem to∫

x; (x)<1

div(ϕ · u) dx

yields ∫

Γ

ϕu · ν do =∫

Γ

ϕ ι∗σ.

Hence ι∗σ = u · ν do =do|∇| , cf. (2.1.2).

In the sequel, we shall make use of “polar coordinates with respect to ”. By this weunderstand the diffeomorphism

H : R+ × Γ −→ Rn \ 0 : (t, ω) 7−→ ta ω

with the inverse

H−1 : Rn \ 0 −→ R+ × Γ : x 7−→((x), (x)−a x

).

From

H∗(dx1 ∧ · · · ∧ dxn) = (a1ta1−1ω1dt+ ta1dω1) ∧ . . .

= t|a| dω1 ∧ · · · ∧ dωn︸︷︷︸=0

+t|a|−1dt ∧ ι∗σ

(comp. Krée [1], pp. 8, 36; Grudzinski [1], Prop. 1.86, p. 67), we conclude that

∫

Rn

ϕ(x) dx =

∞∫

0

t|a|−1 dt∫

Γ

ϕ(ta ω)σ(ω)

4 1 Preliminaries

for ϕ ∈ D(Rn), or, putting it differently, H is an isomorphism of the measure spaces(R+, t

|a|−1dt) ⊗ (Γ, σ) and (Rn \ 0, dx). We shall in the following write very often σinstead of ι∗σ, and we shall identify the differential forms t|a|−1dt, σ, dx with the positivemeasures they induce.

We denote by D(Γ ) the space of C∞ functions on the compact manifold Γ = −1(1)and we endow it with the topology of uniform convergence with respect to all derivatives,i.e. with the seminorms

D(Γ ) −→ R : ϕ 7−→ max∣∣∂α

(ϕ((x)−a x)

)∣∣; 12≤ (x) ≤ 2, x ∈ Rn

,

α ∈ Nn0 . Then D(Γ ) is a Fréchet space.The Lebesgue spaces Lp(Γ ), 1 ≤ p ≤ ∞, are defined as usual with respect to the

measure corresponding to σ, i.e., the equivalence class of the measurable function f :Γ→C

belongs to Lp(Γ ), 1 ≤ p <∞, iff

‖f‖pp :=∫

Γ

|f(ω)|p σ(ω) <∞.

Since Γ is compact, we have Lp(Γ ) ⊂ L1(Γ ) for 1 ≤ p ≤ ∞. Finally, we denote by D′(Γ )the dual of D(Γ ), and we embed L1(Γ ) into D′(Γ ) by means of σ, i.e.

L1(Γ ) → D′(Γ ) : f 7−→(ϕ 7→

∫

Γ

ϕ(ω)f(ω)σ(ω))=: 〈ϕ, f〉,

cf. Gårding [2], pp. 385. 386. Let us point out that the dual of D(Γ ) actually corresponds tothe space of “distribution densities" or (n − 1)−currents (cf. Schwartz [5], Ch. IX, p. 339;Hörmander [1], p. 145), but that, in our case, this space is canonically identified with thespace of distributions on Γ by means of the measure induced by σ.

Next let us treat the pullback of distributions. If Ω is an open subset of Rn and f is acontinuous, real-valued function on Ω, then we obtain the composition map

f∗ : C(R) −→ C(Ω) : g 7−→ g f.

In order to extend this mapping to D′(R), we have to suppose that f ∈ C∞ and f is regular,i.e. f ′(x) 6= 0 for all x ∈ Ω. Then there exists a unique linear continuous map f∗ :D′(R) −→ D′(Ω), the restriction of which to C(R) → D′(R) is the composition above.More explicitly, for T ∈ D′(R), we have

∀ ϕ ∈ D(Ω) : 〈ϕ, f∗T 〉 = 〈 dds

∫

x∈Ω; f(x)<s

ϕ(x) dx, Ts〉,

cf. Friedlander [1], (7.2.4), p. 82. In fact, if x0 ∈ Ω and ∂if(x0) 6= 0, then we can choosey1 = x1, . . . , yi = f, yi+1 = xi+1, . . . , yn = xn as new coordinates in a neighbourhood U

1.2 Convolution of Measures 5

of x0. By means of a partition of unity we can suppose that ϕ ∈ D(U) and set

ψ(y) = ϕ(x(y))∣∣∂x∂y

∣∣. Then

dds

( ∫

f(x)<s

ϕ(x) dx

)=

dds

( ∫

yi<s

ψ(y) dy

)

=

∫ψ(y1, . . . , s, yi+1, . . . , yn) dy1 . . . dyi . . . dyn =: χ(s)

where χ ∈ D(R). Since the map D(U) −→ D(R) : ϕ 7−→ χ is continuous, the same holdsfor the transposition, which is f∗. In particular, if T ∈ C(R), then

〈ϕ, f∗T 〉 =∫χ(s)T (s) ds =

∫χ(yi)T (yi) dyi

=

∫T (yi)dyi

∫ψ(y) dy1 . . . dyi . . . dyn

=

∫ϕ(x(y)

)∣∣∣∂x∂y

∣∣∣T (yi) dy =

∫

U

ϕ(x)T(f(x)

)dx.

More generally, the pullback can be defined in the same way for any submersive C∞

map between two C∞ manifolds, cf. Hörmander [1], Thm. 6.1.2, p. 134. In particular, ifΓ = −1(1) is as above and f : Γ −→ R is C∞ and regular, we obtain the pullbackf∗ : D′(R) −→ D′(Γ ) where

〈ϕ, f∗T 〉 = 〈 dds

∫

ω∈Γ ; f(ω)<s

ϕ(ω)σ(ω), Ts〉 for ϕ ∈ D(Γ ) and T ∈ D′(R).

Finally note that f∗T is also well-defined if T is continuous at the irregular values of f.

1.2 Convolution of Measures

Following Schwartz [6], exp. 22, and [3], § 5, p. 131, and Horváth [8], we shall define theconvolution of distributions generalizing that of measures. Let us therefore repeat first thatthe space of Radon measures M = M(Rn) can be defined in two equivalent ways: Eitheras (signed) Borel measures which are finite on every compact subset of Rn or as continuouslinear functionals on the locally convex topological vector space K = K(Rn) of continuousfunctions with compact support equipped with the inductive limit topology with respect tothe subspaces (KK , ‖ · ‖∞), K ⊂ Rn compact, where KK = f ∈ K; supp f ⊂ K,cf. Dieudonné [2], Ch. XIII, § 1, p. 96. In particular, integrable measures (often called

“bounded" or “finite" measures) are Radon measures fulfilling∫

Rn

|µ| <∞, or, alternatively,

6 1 Preliminaries

linear functionals on K which are continuous with respect to the topology of uniformconvergence (Bourbaki [3], § 2, 9, p. 54) and hence can be extended to linear and continuousfunctionals on the completion

C0 = C0(Rn) = f ∈ C(Rn); lim|x|→∞

f(x) = 0

of K with respect to ‖ · ‖∞, cf. Dieudonné [2], (13.20.6), p. 196.More generally, a given µ ∈ M1 – where M1 denotes the space of integrable measures

– can be applied to all functions in BC(Rn), the space of bounded, continuous functions(Dieudonné [2], (13.20.5), p. 196).

A remarkable assertion of Schwartz [2], p. 102, shows that M1 coincides with the dualspace of BC if BC equipped with a suitable topology coarser than the topology of uniformconvergence generated by the norm ‖·‖∞.We present this statement and its proof followingBuck [1], Thm. 2, p. 99 (cf. also Rubel and Shields [1], 2.11, p. 243).

Proposition 1.2.1. If BC = f : Rn −→ C; f is continuous and bounded is equipped

with the locally convex topology b generated by the seminorms

BC −→ R : f 7−→ ‖f · g‖∞, g ∈ C0,

then BC′b = M1. The topology b is called the “strict or Buck topology".

Proof. Since the maps

(C0, ‖ · ‖∞) → (BC, ‖ · ‖∞)id−→ BCb

are continuous and C0 is dense in BCb, we directly obtain

BC′b ⊂ (C0, ‖ · ‖∞)′ = M1.

In order to show the reverse inclusion, take µ ∈ M1 and define F : BC −→ C : f 7−→∫f µ. Evidently, F is well-defined and linear. The continuity of F with respect to the strict

topology is implied, once we have constructed g ∈ C0(Rn) with g(x) > 0 for all x ∈ Rn

andµ

g∈ M1 and thus shown M1 = C0 · M1. In fact, then

|F (f)| =∣∣∣∫(f · g)µ

g

∣∣∣ ≤ ‖f · g‖∞ ·∫ |µ|

g.

If µ has compact support, thenµ

g∈ M1 for any g ∈ C0 which equals 1 on suppµ. Let us

therefore suppose that suppµ is not compact. Let χ : [0,∞) 7−→ [0, 1] be continuous andnon-decreasing and fulfill

χ(t) =

0 : t ≤ 1/2,

1 : t ≥ 1.


If we define

g(x) :=

(∫χ( |ξ||x|

)|µ|(ξ)

)1/2

,

then g is continuous, by Lebesgue’s theorem, and tends to 0 for |x| → ∞. Furthermore,g is radially symmetric and it is positive for all x since suppµ is not compact. Define thepositive measure ν ∈ M1([0,∞)) by 〈ψ, ν〉 = 〈ψ(|x|), |µ|〉 for ψ ∈ C0([0,∞)) and thenon-increasing function h by h(r) := ν([r,∞)) for r > 0. Then ν coincides with theRiemann-Stieltjes measure −dh. Moreover,

g(x)2 =

∫χ( |ξ||x|

)|µ|(ξ) =

∫χ( t

|x|)ν(t) ≥ ν([|x|,∞)) = h(|x|),

and hence, finally,

∫ |µ|g

≤∫ |µ|(x)√

h(|x|)= 〈 1√

h, ν〉 = −

∞∫

0

dh()√h()

=

h(0)∫

0

du√u= 2

√h(0) = 2

(∫|µ|

)1/2

<∞.

The proof is complete.

Remarks. 1) It can be shown that the topology of BCb, i.e. the strict topology, is the finestlocally convex topology which induces on the norm balls f ∈ BC; ‖f‖∞ ≤ N, N ∈ N,the compact-open topology, i.e. the topology of uniform convergence on compact subsetsof Rn, cf. Dorroh [1]; Cooper [1], Prop. C, p. 589, and Prop. 3, p. 590; Collins [1], p. 211.2) If the sequence ηkk∈N ⊂ C0 is an “approximate unit", i.e., the ηk are uniformlybounded and converge to 1 in the compact-open topology, then also ηk → 1 in BCb fork → ∞, and thus

∫µ = BCb

〈1, µ〉M1 = limk→∞C0〈ηk, µ〉M1 = lim

k→∞

∫ηk(x)µ(x).

Definition 1.2.2. The Radon measures µ, ν ∈ M(Rn) are called convolvable iff

∀ ϕ ∈ K(Rn) : ϕ∆(µ⊗ ν) ∈ M1(R2n),

where ϕ∆ ∈ BC(R2n) is defined by ϕ∆(x, y) = ϕ(x + y) for x, y ∈ Rn. If µ, ν are

convolvable, then their convolution µ ∗ ν ∈ M(Rn) is defined by

〈ϕ, µ ∗ ν〉 =∫ϕ∆(µ⊗ ν) = BCb(R2n)〈1, ϕ∆(µ⊗ ν)〉M1(R2n), ϕ ∈ K(Rn).

8 1 Preliminaries

Remarks. 1) Note that µ ∗ ν is continuous with respect to the inductive limit topology onK (and hence µ ∗ ν ∈ M), since

|〈ϕ, µ ∗ ν〉| =∣∣∣∫ϕ∆ψ∆ · µ⊗ ν

∣∣∣ ≤ ‖ϕ‖∞∫

|ψ∆||µ⊗ ν|

if ϕ has its support in a fixed compact setK ⊂ Rn and ψ ∈ K is chosen such that ψ(x) = 1for x ∈ K.2) The definition of convolution as above goes back to Bourbaki: “ϕ is integrable for µ ∗ νiff ϕ∆ is integrable for µ ⊗ ν”, cf. Bourbaki [4], Ch. VIII, § 1, p. 121; Dieudonné [2],Ch. XIV, § 5, p. 246. The reformulation in Def. 1.2.2 is due to Horváth [8], p. 184. In fact,ϕ∆ is integrable for µ ⊗ ν iff ϕ∆(µ ⊗ ν) ∈ M1, cf. Bourbaki [2], § 5, Cor. du Thm. 1,p. 48; Dieudonné [2], (13.20.3), p. 195.3) If, as always, we consider L1

loc(Rn) as a subspace of M by identifying f with f(x)dx,

thensinx

x6∈ M1(R1) and eix2 6∈ M1(R1). Nevertheless, it is possible to integrate these

measures, e.g. by defining∫S := FS(0) if S ∈ S ′(Rn) and FS is continuous at 0,

cf. Ricards and Youn [1], p. 114, Def. (a). On account of this definition, we have in theexamples above

∫sinx

x= πY (1− |x|)|x=0 = π and

∫eix2

=√πei(π−x2)/4|x=0 =

√πeiπ/4,

which values coincide with the well-known values of the corresponding improper definiteintegrals.

Let us note that, in the sense of the above definition, we also have∫xn sinx = 0 for n ∈ N

though the corresponding improper integrals diverge.4) If µ, ν are convolvable measures and f, g ∈ BC, then fµ, gν are convolvable as well.This fact shows at once that eia|x|2 and eib|x|2 , a, b ∈ R, are not convolvable as measures.Note that this is also implied by the equivalence

µ, ν convolvable ⇐⇒ |µ|, |ν| convolvable.

(Let us point out that, nevertheless, eia|x|2 and eib|x|2 are convolvable as distributions for(a, b) ∈ R2 \ 0, cf. Wagner [3], Satz 4, p. 473.)

Concerning the regularization of measures, i.e. the convolution of a measure with acontinuous function with compact support, we state the following (cf. Horváth [4], Ch. 4,§ 10, Prop. 1 and ex. 5, pp. 402, 407).

Lemma 1.2.3. 1) If ϕ ∈ K and µ ∈ M, then ϕ, µ are convolvable (as measures) and ϕ ∗µis a continuous function given by

(ϕ ∗ µ)(x) = K〈ϕ(x− y), µy〉M =

∫ϕ(x− y)µ(y).


2) The bilinear map

K ×M −→ Cc : (ϕ, µ) 7−→ ϕ ∗ µis hypocontinuous. (Here Cc denotes the space of continuous functions on Rn with the

compact-open topology.) If K is a compact subset of Rn, then the map

KK ×M −→ Cc : (ϕ, µ) 7−→ ϕ ∗ µ

is continuous.

Proof. 1) For ψ ∈ K, |ψ| ∗ |ϕ| ∈ K and hence, by Fubini’s theorem,∫ ∣∣ψ∆(ϕ⊗ µ)

∣∣ =∫ (∫

|ψ(x+ y)ϕ(x)| dx)|µ|(y) = 〈(|ψ| ∗ |ϕ|), |µ|〉

is finite and thus ϕ, µ are convolvable. By the same reason,

〈ψ,ϕ ∗ µ〉 =∫ (∫

ψ(x+ y)µ(y))ϕ(x) dx

=

∫ψ(z)

(∫ϕ(z − y)µ(y)

)dz.

2) The hypocontinuity of ∗ means that this mapping is separately continuous, uniformlywith respect to bounded sets in the “fixed" variable. First, if B ⊂ K is a bounded set, thenthere exists a compact subset K of Rn and a constant C > 0 such that suppϕ ⊂ K and‖ϕ‖∞ ≤ C for ϕ ∈ B. Hence, for L ⊂ Rn compact, µ ∈ M, ϕ ∈ B, we obtain

‖(ϕ ∗ µ)|L‖∞ = supx∈L

∣∣∣∫ϕ(x− y)µ(y)

∣∣∣

= supx∈L

∣∣∣∫

L−K

ϕ(x− y)µ(y)∣∣∣ ≤ C

∫

L−K

|µ|.

This yields the continuity of µ 7−→ ϕ ∗ µ uniformly with respect to ϕ ∈ B, since µ 7−→∫

L−K

|µ| is a continuous seminorm on M.

Second, if µ belongs to a bounded set B of M and K,L are compact subsets of Rn, then∫

L−K

|µ| is bounded by a constant CK,L for µ ∈ B. Thus ‖(ϕ ∗ µ)|L‖∞ ≤ CK,L‖ϕ‖∞ if

ϕ ∈ KK . This estimate implies the continuity of ϕ 7−→ ϕ ∗ µ, uniformly with respect toµ ∈ B.

Remark. For distributions, the convolvability can also be characterized by various otherconditions (see Prop. 1.3.4), which, however, for measures are not equivalent.

10 1 Preliminaries

Proposition 1.2.4. For two measures µ, ν ∈ M, we have the implications

(i) =⇒ (ii), (i) =⇒ (iii), (i) ⇐⇒ (iv) where

(i) µ, ν are convolvable;

(ii) ∀ ϕ ∈ K : (ϕ ∗ µ)ν ∈ M1(Rn);(iii) ∀ ϕ, ψ ∈ K : (ϕ ∗ µ)(ψ ∗ ν) ∈ L1(Rn);(iv) ∀ K ⊂ Rn compact : |µ⊗ ν|(K∆) <∞ where K∆ = (x, y) ∈ R2n ; x+ y ∈ K.If (i) holds, then the convolution µ ∗ ν is also given by

〈ϕ, µ ∗ ν〉 = BCb(Rn)〈1, (ϕ ∗ µ)ν〉M1(Rn), ϕ ∈ K(Rn).

Proof. Evidently, (i) and (iv) are equivalent, since∫

R2n

|ϕ∆||µ⊗ ν| ≤ ‖ϕ‖∞ · |µ⊗ ν|(K∆)

ifK = suppϕ, ϕ ∈ K, and conversely, forK ⊂ Rn compact, |µ⊗ν|(K∆) ≤∫

|ϕ∆||µ⊗ν|if ϕ ∈ K such that ϕ = 1 on K.The implication (i) =⇒ (ii) follows from Fubini’s theorem similarly to the proof of the firstpart of Lemma 1.2.3, cf. also Dieudonné [2], (14.9.4), p. 257. This likewise yields

〈ϕ, µ ∗ ν〉 = BCb〈1, (ϕ ∗ µ)ν〉M1 , ϕ ∈ K.

Let us finally show (i) =⇒ (iii) : Using Fubini’s theorem we have, for ϕ,ψ ∈ K,∫

|(ϕ ∗ µ)(x)| · |(ψ ∗ ν)(x)| dx ≤∫ (∫

Rnx

(|ϕ| ∗ |µ|)(x) · |ν|(x− y))|ψ(y)| dy

=

∫ (∫ (|τ−yϕ| ∗ |µ|

)(z)|ν|(z)

)|ψ(y)| dy

=

∫〈|τ−yϕ|, |µ| ∗ |ν|〉 |ψ(y)| dy ≤ CK‖ϕ‖∞‖ψ‖1 <∞

if τ−yϕ(x) = ϕ(x+y), K = x ∈ Rn; x+y ∈ suppϕ, y ∈ suppψ andCK =

∫

K

|µ|∗|ν|.

Hence (ϕ ∗ µ)(ψ ∗ ν) ∈ L1.

Example 1.2.5. In order to see that the implications (i) =⇒ (ii) and (i) =⇒ (iii) cannot bereversed, we can take

µ =eix2

x+ i, ν = 1 ∈ M(R1).

Then µ, ν are not convolvable, since

1.3 Convolution of Distributions 11

∫

R2

∫ ∣∣∣ϕ(x+ y)eix2

x+ i

∣∣∣dx dy = ‖ϕ‖1∫

dx√1 + x2

= ∞

if ϕ ∈ K(R1) \ 0. On the other hand, ϕ ∗ µ ∈ L1(R1), since

(i − x)(ϕ ∗ µ) = −(xϕ) ∗ µ+ ϕ ∗((i − x)µ

)∈ K ∗ L2 + ϕ ∗ eix2

⊂ L1 ∗ L2 + eix2F(ϕ eix2)

(2x)

⊂ L2 + L2 = L2,

and thus ϕ ∗ µ ∈ 1

i − x· L2 ⊂ L1.

1.3 Convolution of Distributions

In a similar way as D′ generalizes the space M of Radon measures, the space D′L1 of

“integrable distributions" is a generalization of the space M1 of integrable measures.D′L1(Rn) was defined by L. Schwartz as the dual of

.B =

.B(Rn) = ϕ ∈ E(Rn); ∀ α ∈ Nn0 : ∂αϕ ∈ C0.

.B is a Fréchet space under the seminorms ϕ 7−→ ‖∂αϕ‖∞, α ∈ Nn0 , and D is denselycontained in

.B.

Similarly as in the measure-theoretic case, Schwartz [2], p. 100, and [5], p. 203, showedthat also D′

L1 = B′c, where B = DL∞(Rn) is the space of infinitely differentiable functions

with all derivatives bounded, and the subscript c indicates that B is equipped with the finestlocally convex topology inducing the topology of E on the subsets of B which are boundedwith respect to its Fréchet topology generated by the seminorms ϕ 7−→ ‖∂αϕ‖∞, α ∈ Nn0 .(Note that this topology is coarser than the Fréchet topology on B = DL∞ .) Similarly toProp. 1.2.1, let us define explicitly a locally convex topology b on B such that B′

b = D′L1 .

(Actually, Bb = Bc, cf. Dierolf and Dierolf [1], (3.5) Cor. (a), p. 71.)

Proposition 1.3.1. If B = DL∞ is equipped with the locally convex topology b generated

by the seminorms

B −→ R : ϕ 7−→ ‖g · ∂αϕ‖∞, g ∈ C0, α ∈ Nn0 ,

then B′b = D′

L1 .

Proof. Since the maps.B → B id−→ Bb

are continuous and.B is dense in Bb, we directly obtain B′

b ⊂.B′ = D′

L1 . In order toshow the reverse inclusion, take T ∈ D′

L1 and use the representation theorem for integrabledistributions in Schwartz [5], Ch. VI, Thm. XXV, p. 201:

12 1 Preliminaries

T =∑

|α|≤m∂αfα with fα ∈ L1 ⊂ M1.

Due to Prop. 1.2.1, fα ∈ BC′b, i.e.,

∃ gα ∈ C0 : ∀ f ∈ BC :∣∣∣∫f · fα dx

∣∣∣ ≤ ‖f · gα‖∞,

and hence, for ϕ ∈.B,

|〈ϕ, T 〉| ≤∑

|α|≤m|〈ϕ, ∂αfα〉| =

∑

|α|≤m

∣∣∣∫∂αϕ · fα dx

∣∣∣ ≤∑

|α|≤m‖gα · ∂αϕ‖∞.

Thus T ∈ B′b and the proof is complete.

Remark. Similarly as in the measure-theoretic case, the “integral" of T ∈ D′L1 can be

approximated asBc〈1, T 〉D′

L1= limk→∞

.B〈ηk, T 〉D′

L1

if ηk → 1 in Bc as k → ∞, i.e., the functions ∂αηk are uniformly bounded for every α, andηk → 1 in E , cf. Schwartz [6], exp. 22; Dierolf and Voigt [1], (1.1), Prop. (c), p. 187. In thelast reference, it is also shown that the convergence of 〈ηk, T 〉 for T ∈ D′ and for sequencesηk as above implies T ∈ D′

L1 .

The following definition of the convolution of distributions based on the space D′L1 goes

back to Schwartz [6], exp. 22, and [3], p. 131; Horváth [8], (2) Déf., p. 185.

Definition 1.3.2. The distributions S, T ∈ D′(Rn) are called convolvable iff

∀ ϕ ∈ D(Rn) : ϕ∆(S ⊗ T ) ∈ D′L1(R2n),

where, as above, ϕ∆ ∈ DL∞(R2n) is defined by ϕ∆(x, y) = ϕ(x + y) for x, y ∈ Rn. If

S, T are convolvable, then their convolution S ∗ T ∈ D′(Rn) is defined by

〈ϕ, S ∗ T 〉 = Bb(R2n)〈1, ϕ∆(S ⊗ T )〉D′

L1 (R2n), ϕ ∈ D(Rn).

Remarks. 1) If, instead of D′L1 , a larger normal space E of distributions which still

contains the function 1 in its dual is used, then more general notions of convolvability arise.An example of such a space was constructed in Dierolf and Voigt [2], p. 84, by taking

E = (.Bi)′ =

∑

|α|≤m∂α

((1 + |x|2)|α|/2µα

); µα ∈ M1

,

which is the dual of the Fréchet space

.Bi =

ϕ ∈

.B; ∀ α ∈ Nn0 : (1 + |x|2)|α|/2∂αϕ ∈ C0

.


In R1, we then havesinx

x∈ (

.Bi)′ \ D′

L1 and hence 1 andsinx

xare convolvable in this

sense (with 1 ∗ sinx

x= π), but not in the sense of Def. 1.3.2, cf. Dierolf and Voigt [2],

(5.7), p. 84.2) A different approach to generalizing the concept of convolution is based on the observationthat FU is continuous for U ∈ D′

L1 and hence 〈ϕ, S ∗T 〉 is just the value of F(ϕ∆(S⊗T )

)

at 0. This is used in Ricards and Youn [1], pp. 114, 123, where S, T ∈ S ′(Rn) arecalled convolvable if, for all ϕ ∈ D(Rn), F

(ϕ∆(S ⊗ T )

)is a continuous function in a

neighbourhood of 0 ∈ R2n. E.g., then the distributions 1 andsinx

xare convolvable in this

sense as well.

Concerning the convolution of a distribution with a test function, we have the following(cf. Schwartz [5], Ch. VI, Thm. XII, p. 167):

Lemma 1.3.3. 1) If ϕ ∈ D and T ∈ D′, then ϕ, T are convolvable and ϕ ∗ T ∈ E is given

by (ϕ ∗ T )(x) = D〈ϕ(x− y), Ty〉D′ .2) The bilinear mapping D ×D′ −→ E : (ϕ, T ) 7−→ ϕ ∗ T is hypocontinuous.

In contrast to the measure-theoretic case, the convolvability of distributions can also becharacterized by the conditions analogous to (ii), (iii), (iv) in Prop. 1.2.4, cf. Shiraishi [1],Thm. 2, p. 24; Horváth [9], Prop. 1, p. 185.

Proposition 1.3.4. For two distributions S, T, the following five conditions are equivalent:

(i) S, T are convolvable;

(ii) ∀ ϕ ∈ D : (ϕ ∗ S)T ∈ D′L1(Rn);

(iii) ∀ ϕ,ψ ∈ D : (ϕ ∗ S)(ψ ∗ T ) ∈ L1(Rn);(iv) ∀ a > 0 : ∃ C > 0 : ∃ m ∈ N :

∀ ψ ∈ D(R2n) with suppψ ⊂ ∆a := (x, y) ∈ R2n; |x+ y| ≤ a :

|〈ψ, S ⊗ T 〉| ≤ C∑

|α|≤m‖∂αψ‖∞;

(v) S ⊗ T ∈ (.B∆)′, where

.

B∆ is the strict inductive limit of.Bk := ψ ∈

.B(R2n); suppψ ⊂ ∆k, k ∈ N,

equipped with the seminorms ψ 7→ ‖∂αψ‖∞, α ∈ N2n0 .

If (i) holds, then the convolution S ∗ T is also given by

〈ϕ, S ∗ T 〉 = Bb(Rn)〈1, (ϕ ∗ S)T 〉D′

L1 (Rn), ϕ ∈ D(Rn).

Proof. The implication (i) =⇒ (iv) is almost immediate by choosing ϕ ∈ D with ϕ(x) = 1for |x| ≤ a and using ϕ∆(S ⊗ T ) ∈ D′

L1(R2n). Conversely, if ϕ ∈ D(Rn) with supportin x ∈ Rn; |x| ≤ a, then condition (iv) implies that |〈ψ,ϕ∆(S ⊗ T )〉| is bounded byseminorms ‖∂αψ‖∞ of DL∞(R2n) and hence ϕ∆(S ⊗ T ) ∈ D′

L1(R2n). Furthermore, (iv)is equivalent to the continuity of S⊗T considered as a linear functional on the completions

.Ba(R2n) = ψ ∈

.B(R2n); suppψ ⊂ ∆a, a > 0,

14 1 Preliminaries

of the space ψ ∈ D(R2n); suppψ ⊂ ∆a equipped with the seminorms ψ 7→ ‖∂αψ‖∞,α ∈ N2n

0 , (which is the induced topology of DL∞(R2n)), and hence to S ⊗ T ∈ (.B∆)′.

This proves the equivalence of (i), (iv), (v).Let us next show that (v) implies (ii) and (iii). If ϕ,ψ, χ ∈ D(Rn), then simple

calculations show that

〈χ,((ϕ ∗ S)T

)∗ ψ〉 = 〈ϕ(x+ y)(ψ ∗ χ)(y), Sx ⊗ Ty〉

and

〈χ, (ϕ ∗ S) · (ψ ∗ T )〉 = 〈∫ϕ(t+ x)ψ(t− y)χ(t) dt, Sx ⊗ Ty〉.

The two mappingsC0 −→

.B∆ : χ 7−→ ϕ(x+ y)(ψ ∗ χ)(y)

and

C0 −→.B∆ : χ 7−→

∫ϕ(t+ x)ψ(t− y)χ(t) dt

are well-defined, linear and continuous. (E.g. ‖ϕ(x + y)(ψ ∗ χ)(y)‖∞ ≤ Cϕ,ψ‖χ‖∞.) Bycomposing these mappings with S ⊗ T :

.B∆ −→ C we conclude that

∀ ϕ,ψ ∈ D(Rn) : ∃ C > 0 : ∀ χ ∈ C0(Rn) :

|〈χ,((ϕ ∗ S)T

)∗ ψ〉| ≤ C‖χ‖∞ and |〈χ, (ϕ ∗ S) · (ψ ∗ T )〉| ≤ C‖χ‖∞,

and hence((ϕ∗S)T

)∗ψ, (ϕ∗S)·(ψ∗T ) ∈ M1. Since these are continuous functions, they

in fact belong to L1. This yields (iii) and, by Thm. XXV in Schwartz [5], Ch. VI, p. 201,

also (ii). Furthermore, if (i) and (ii) hold and if ψ ∈ D(Rn) with∫ψ(x) dx = 1 and

ψh(x) = ψ(xh

)h−n for h > 0, then ψh → δ in E ′(Rn) for hց 0 and hence ψh ∗ χ→ χ

in D(Rn) for χ ∈ D(Rn). Thus

〈χ, (ϕ ∗ S)T 〉 = limhց0

〈χ,((ϕ ∗ S)T

)∗ ψh〉

= limhց0

〈ϕ(x+ y)(ψh ∗ χ)(y), Sx ⊗ Ty〉

= 〈ϕ(x+ y)χ(y), Sx ⊗ Ty〉 = Bb〈1⊗ χ, ϕ∆(S ⊗ T )〉D′

L1.

Therefore, if χj ∈ D(Rn) and χj → 1 in Bb(Rn), then also 1 ⊗ χj → 1 in Bb(R2n) andthus

〈ϕ, S ∗ T 〉 = Bb(R2n)〈1, ϕ∆(S ⊗ T )〉D′

L1 (R2n) = Bb(Rn)〈1, (ϕ ∗ S)T 〉D′

L1 (Rn).

Conversely, if (ii) holds and ϕ ∈ D(Rn) and K ⊂ Rn is compact with suppϕ ⊂K, then

the form

G : DK(Rn)×.B(Rn) −→ C : (ψ, χ) 7−→ .

B〈χ,((ψϕ) ∗ S

)T 〉D′

L1


is well-defined and bilinear. G is also separately continuous, trivially with respect to χ, and,by the closed graph theorem applied to

DK −→ D′L1 : ψ 7−→

((ψϕ) ∗ S

)T,

also with respect to ψ. Since DK and.B are Fréchet spaces, G is continuous, cf. Robertson

and Robertson [1], Ch. VII, Prop. 11, p. 136. Therefore G extends to a linear continuousmap G on the completion DK⊗π

.B = DK⊗ε

.B of the tensor product. (Note that DK is

nuclear, cf. Treves [1], Cor. to Thm. 51.5, p. 530.)If ζ ∈ DK(Rn) and ζ = 1 on suppϕ, then F :

.B −→ DK : ψ 7−→ ψ · ζ is continuous,

and hence the same holds for G defined by

G :.B(R2n) ≃

.B(Rn)⊗ε

.B(Rn) F ⊗εId−→ DK⊗ε

.B G−→ C,

cf. Treves [1], Def. 43.6, p. 439, and Schwartz [3], Prop. 17, p. 59.

For ψ, χ ∈ D, we have

G(ψ ⊗ χ) = G(ψ · ζ ⊗ χ) = G(ψ · ζ, χ) = D〈χ(y)ψ(x+ y), ϕ∆(S ⊗ T )〉D′ ,

and hence generally,

∀ h ∈ D(R2n) : G(h) = D〈h(x+ y, y), ϕ∆(S ⊗ T )〉D′ .

Viewing G as an integrable distribution, i.e. as a continuous linear functional on.B(R2n) we

infer thatϕ∆(S ⊗ T ) = G(x+ y, y) ∈ D′

L1(R2n)

and thus the validity of condition (i).Let us last show (iii) =⇒ (i). For compact subsets K ⊂ Rn, the hypothesis (iii) yields

thatH : DK ×DK −→ M1 : (ϕ,ψ) 7−→ (ϕ ∗ S)(ψ ∗ T )

is well-defined and bilinear and, furthermore, by the closed graph theorem, separatelycontinuous. Since DK is a Fréchet space, H is continuous, cf. Treves [1], p. 354, Corollary.Therefore the mapping

DK ×DK ×.B −→ C : (ϕ,ψ, χ) 7−→ 〈χ,H(ϕ,ψ)〉

is also continuous and can be defined on the completed tensor product of the spacesDK ,DK ,

.B. Hence there exist m ∈ N and C > 0 such that for f ∈ DK⊗DK⊗

.B we

have ∣∣〈∫f(t+ x, t− y, t) dt, S ⊗ T 〉

∣∣ ≤ C∑

α∈N3n0 ,|α|≤m

‖∂αf‖∞.

Let us choose h ∈ D(Rn) with∫h(t) dt = 1 and set

16 1 Preliminaries

f(x, y, t) := ϕ(x− y)χ(x− t, t− y)h(x)

for fixed ϕ ∈ D(Rn), χ ∈ D(R2n). Then∫f(t+ x, t− y, t) dt = ϕ∆ · χ and

supp f ⊂ K ×K × Rn with K depending only on ϕ and h but not on χ. Thus

∣∣〈χ, ϕ∆(S ⊗ T )〉∣∣ ≤ C

∑

α∈N3n0 ,|α|≤m

‖∂αf‖∞ ≤ C1

∑

α∈N2n0 ,|α|≤m

‖∂αχ‖∞,

where C1 depends on ϕ, h. Hence ϕ∆(S ⊗ T ) ∈ D′L1 , i.e. condition (i) is satisfied.

Remarks. 1) By definition, S, T are convolvable iff T, S are convolvable and hencecondition (ii) is equivalent with

(ii)′ : ∀ ϕ ∈ D(Rn) : S · (ϕ ∗ T ) ∈ D′L1(Rn).

2) Already in 1950, Chevalley [1], p. 112, used condition (iii) to define the convolution ofdistributions in a symmetric manner. Conditions (i), (ii) were first given in Schwartz [6],exp. 22, pp. 1, 2. He also showed the implication (i) =⇒ (ii). We observe that condition(ii) was given in a somewhat hidden form in Chevalley [1], p. 68, and was formulated as(ii) in Shiraishi [1], p. 19. The equivalence of (i), (ii), (iii) was proven also in Shiraishi [1],cf. Thm. 2, p. 24. Later on Roider [1], pp. 194, 195, gave a different proof of (i) ⇐⇒ (ii).Condition (i) was reinvented in Horváth [8] and formulated as in (iv) in Horváth [9], Prop. 1,p. 185. Our proof of the equivalence of (i), (ii), and (iii) differs from the proofs given byShiraishi and Roider, respectively.3) The equivalence of (ii) and (v) can also be shown directly, by using the isomorphisms

(.B∆)′ ∼−→ D′⊗D′

L1 : U 7−→ U(x− y, y).

and D′⊗D′L1 ≃ L(D;D′

L1), cf. Treves [1], p. 533. Hence, by the closed graph theorem,S ⊗ T ∈ (

.B∆)′ iff ∀ ϕ ∈ D : (ϕ ∗ S)T = (S ⊗ T )(x− y, y)(ϕ) ∈ D′

L1(Rny ).Note that condition (ii) can also be formulated by means of the notion “partiellement

sommable" in Schwartz [3], p. 130, cf. Dierolf and Voigt [1], Thm. (1.3), p. 190, and Horváth[10], p. 8-09. This observation is the starting point for the proof of the equivalence of (i) and(ii) in Roider [1], pp. 194, 195.4) We point out that essentially the same proof as that of Prop. 1.3.4 also furnishes theequivalence of the following conditions on S, T ∈ D′(Rn) for 1 < p ≤ ∞ :(i)’ ∀ ϕ ∈ D : ϕ∆(S ⊗ T ) ∈ D′

Lp(R2n);(ii)’ ∀ ϕ ∈ D : (ϕ ∗ S)T ∈ D′

Lp(Rn);(iii)’ ∀ ϕ,ψ ∈ D : (ϕ ∗ S)(ψ ∗ T ) ∈ Lp(Rn);(iv)’ ∀ a > 0 : ∃ C > 0 : ∃ m ∈ N :

∀ ψ ∈ D(R2n) with suppψ ⊂ ∆a := (x, y) ∈ R2n; |x+ y| ≤ a :

|〈ψ, S ⊗ T 〉| ≤ C∑

|α|≤m‖∂αψ‖q, where

1

p+

1

q= 1;

(v)’ S ⊗ T ∈ (DLq,∆)′, where

1

p+

1

q= 1 and DLq,∆ is the strict inductive limit of


DLq,k := ψ ∈ DLq (R2n); suppψ ⊂ ∆k, k ∈ N, as subspaces of DLq .

Note that, in proving (ii)’ =⇒ (i)’ analogously to the proof of (ii) =⇒ (i), G has to beconstructed slightly different, since

DLq (R2n) 6= DLq (Rn)⊗εDLq(Rn) and DLq (R2n) 6= DLq (Rn)⊗πDLq (Rn)

if 1 < q <∞. Instead, we use the fact that the map

DLq (R2n) −→ DK⊗DLq(Rn) : h 7−→ ζ(x)h(x, y)

is continuous, since it is the composition of the maps

DLq(R2n) −→ C∞(Rn;DLq(Rn)) ≃ C∞(Rn)⊗DLq (Rn) : h 7−→ (x 7−→ h(x,−))

andC∞(Rn)⊗DLq (Rn)

ζ⊗id−→ DK⊗DLq (Rn) : h(x, y) 7−→ ζ(x)h(x, y).

For later use let us add a lemma on the associativity of the convolution.

Lemma 1.3.5. If S, T are convolvable and U ∈ E ′ then U ∗ S and T are convolvable and

(U ∗ S) ∗ T = U ∗ (S ∗ T ).Proof. If ψ1, ψ2 ∈ D, then

(ψ1 ∗ (U ∗ S ) ) · (ψ2 ∗ T ) =((ψ1 ∗ U) ∗ S

)· (ψ2 ∗ T ) ∈ L1(Rn)

by condition (iii) in Prop. 1.3.4, and hence U ∗ S, T are convolvable by the very samecondition.Eventually, for ψ ∈ D, we have

〈ψ,U ∗ (S ∗ T )〉 = 〈U ∗ ψ, S ∗ T 〉 = 〈1, (U ∗ ψ ∗ S)T 〉= 〈1, (ψ ∗ (U ∗ S ) )T 〉 = 〈ψ, (U ∗ S) ∗ T 〉.

Remark. For the general result on the associativity of the convolution of three distributions,cf. Shiraishi [1], Lemma 1, p. 28, and Roider [1], p. 195. The special case of U = ∂jδ isProp. 18 in Horváth [8], p. 192.

In 3.4, we shall use the following regularity result (cf. Ortner and Wagner [1], Prop. 5,p. 362) in the case of p = 1.

Lemma 1.3.6. If 1 ≤ p ≤ ∞ and T ∈ D′Lp(Rn) with ∂jT ∈ Lp(Rn), j = 1, . . . , n, then

T ∈ Lp(Rn).

Proof. We apply L. Schwartz’ parametrix method (Schwartz [5], (VI,6;22), p. 191):If γ ∈ D, γ = 1 on a neighbourhood of 0 and E is a fundamental solution of ∆n, then thereexists ϕ ∈ D such that ∆n

((γE) ∗ T

)− ϕ ∗ T = T.

18 1 Preliminaries

Since T ∈ D′Lp we have ϕ∗T ∈ Lp; on the other hand∆n

((γE)∗T

)=

n∑

j=1

∂j(γE)∗∂jT,

and ∂jT ∈ Lp implies ∂j(γE) ∗ ∂jT ∈ L1 ∗ Lp ⊂ Lp, since ∃ C > 0 : ∀ x ∈ Rn :|∂j(γE)(x)| ≤ C|x|1−n.

1.4 Continuity of Bilinear Mappings

Let us recall that a locally convex topological vector space E is barrelled iff every lowersemicontinuous seminorm on E is continuous. (In fact, this is equivalent with the usualdefinition that every barrel is a neighbourhood of 0.) For the sake of completeness, let usgive a proof of the hypocontinuity of bilinear separately continuous mappings, cf. Horváth[4], Ch. 4, § 7, Thm. 2, p. 360; Treves [1], Thm. 41.2, p. 424.

Proposition 1.4.1. If E,F,G are locally convex spaces, E is barrelled and

u : E × F −→ G is bilinear and separately continuous, then u is hypocontinuous with

respect to the set of bounded subsets of F.

Proof. Let r be a continuous seminorm onG andB a bounded subset of F. ThenE → R+ :x 7−→ r

(u(x, y)

)is a continuous seminorm for fixed y ∈ F. Therefore, by the boundedness

of B,E −→ R+ : x 7−→ sup

y∈Br(u(x, y)

),

is a well-defined lower semicontinuous seminorm, and hence it is continuous due to thebarrelledness of E. This means the hypocontinuity in question.

In order to infer continuity we restrict the class of spaces involved, cf. Grothendieck [2],p. 63, [3], p. 301:

Definition 1.4.2. A locally convex space E is a barrelled DF-space if it is barrelled and

there exists a countable fundamental system of bounded subsets.

The main result is the following, cf. Grothendieck [3], Cor. 1, p. 308.

Proposition 1.4.3. LetE and F be barrelled DF−spaces andG be a locally convex space,

u : E × F −→ G a bilinear, separately continuous mapping. Then u is continuous.

Proof. First we conclude from Proposition 1.4.1 that u is hypocontinuous with respect tothe sets of bounded subsets in E as well as in F. Hence, if r is a continuous seminorm onG and Bj ; j ∈ N is a fundamental system of bounded sets in F, then

pj : E −→ R : x 7−→ supy∈Bj

r(u(x, y)

)

are continuous seminorms onE. By Lemma 1.4.4 below, there exist a continuous seminormp on E and numbers Cj > 0 such that ∀j ∈ N : pj ≤ Cjp. Therefore,

q : F −→ R : y 7−→ supp(x)≤1

r(u(x, y)

),

1.5 Vector-Valued Holomorphic Functions 19

is a well-defined lower semicontinuous seminorm, which is continuous due to the barrelled-ness of F. This yields the inequality

∀x ∈ E : ∀y ∈ F : r(u(x, y)

)≤ p(x)q(y),

which implies the continuity of u.

The following lemma (Grothendieck [2], Lemme 2, p. 64) relies on “Mackey’s countabilitycondition” (cf. Horváth [4], Ch. 2, § 6, Prop. 3, p. 116; Treves [1], Lemma 41.1, p. 422).

Lemma 1.4.4. If E is a barrelled DF−space and pj , j ∈ N, are continuous seminorms,

then there exist a continuous seminorm p and positive numbers Cj such that pj ≤ Cjp for

all j ∈ N.

Proof. Let B1 ⊂ B2 ⊂ · · · be a fundamental system of bounded subsets of E and putCj := 1 + sup

x∈Bj

pj(x). If x ∈ E and i ∈ N is such that x ∈ Bi, then x ∈ Bj for j ≥ i,

and therefore the sequence1

Cjpj(x), j ∈ N, is bounded. Hence p(x) := sup

j∈N

1

Cjpj(x) is

well-defined, and – being a lower semicontinuous seminorm – it is continuous, due to thebarrelledness of E.

The main application of Prop. 1.4.3 occurs when E and F are strong duals of metrizablelocally convex spaces, cf. Grothendieck [3], Thm. 1, p. 302:

Proposition 1.4.5 If G is a metrizable locally convex space and E = G′ is barrelled with

the strong topology, then E is a barrelled DF−space.

Proof. By the metrizability of G, there exists a fundamental system Uj , j ∈ N, of zero-neighbourhoods in G. Since strongly bounded subsets of E = G′ are equicontinuous (Gis infrabarrelled, cf. Horváth [4], Ch. 3, § 6, Prop. 6, p. 217, and § 7, Prop. 3, p. 222), weconclude that the polars Bj = U

j yield a fundamental system of bounded subsets in E.

1.5 Vector-Valued Holomorphic Functions

If U is a topological space andE is a separated locally convex space (i.e. a Hausdorff locallyconvex topological vector space), then a function f : U −→ E is called weakly continuous

if g f is a continuous function for each g ∈ E′ (cf. Bourbaki [1], Ch. III, § 1, no. 1, p. 74).In general, a weakly continuous function does not have to be continuous. E.g., the

standard basis vectors ej ∈ l2, j ∈ N, tend weakly to 0, but not strongly, and hence

f :

1

j; j ∈ N

∪ 0 −→ l2

x 7−→ej : x = 1/j,

0 : x = 0.

20 1 Preliminaries

is weakly continuous, but not continuous. (Note that, in contrast, for Montel spaces, thesetwo concepts of continuity agree – at least for metrizable U – since then weakly convergingsequences converge uniformly on bounded subsets ofE′, cf. Treves [1], Prop. 36.11, p. 377.)

If E is quasi-complete (i.e. each bounded closed subset of E is complete), U ⊂ Rn

is open and f : U −→ E is weakly C∞ (i.e. f ∈ E(U)⊗E, then f is also strongly C∞

(cf. Schwartz [2], Lemme II, p. 146; Horváth [7], Lema (1.2.1), p. 60). Analogously, theconcepts of strong and weak holomorphy agree in quasi-complete spaces (cf. Grothendieck[1], Thm. 1, p. 37; Horváth [7], Teorema 1.1.4, p. 57; Edwards [1], 8.14.7, p. 563 ff.;Prop. 1.5.2 below). Let us mention that quasi-complete locally convex spaces arise naturallyas dual spaces F ′ with the topology of uniform convergence on a collection of boundedsubsets of F (cf. Horváth [4], Ch. 3, § 6, Thm. 1, p. 218).

Definition 1.5.1. Let E be a Hausdorff locally convex topological vector space and U ⊂ C

open. Then f : U −→ E is called (strongly) holomorphic iff f is complex differentiable for

all z0 ∈ U, i.e. limz→z0

f(z)− f(z0)

z − z0exists inE. The function f is called weakly holomorphic

iff g f : U −→ C is holomorphic for all g ∈ E′ i.e. f ∈ H(U)⊗E.

Proposition 1.5.2. Let U ⊂ C open and E be a Hausdorff quasi-complete locally convex

space. Then f : U −→ E is weakly holomorphic if and only if it is strongly holomorphic.

Proof. Trivially, the existence of the strong derivatives limz→z0

f(z)− f(z0)

z − z0implies the weak

holomorphy of f, i.e. the existence of

limz→z0

⟨f(z)− f(z0)

z − z0, g

⟩= limz→z0

g f(z)− g f(z0)z − z0

for g ∈ E′ and z0 ∈ U.

Conversely, suppose that f is weakly holomorphic. Then f is continuous, since, for

z0 ∈ U,f(z)− f(z0)

z − z0is weakly bounded for all z sufficiently near z0, and hence it is

also strongly bounded by Mackey’s theorem (cf. Horváth [4], Ch. 3, § 5, Thm. 3, p. 209;Robertson and Robertson [1], Ch. IV, § 1, Thm. 1, p. 67) which implies that f(z) → f(z0)in E for z → z0. Therefore, and since E is quasi-complete, we can define the line integral(Bourbaki [1], Ch. III, § 3, no. 1, Def. 1, p. 75, and no. 3, Cor. 2, p. 80)

f1(z) :=1

2πi

∫

|w−z0|=r

f(w)

w − zdw for |z − z0| < r

if 0 < r < dist (z0, ∂U) is fixed. Since we have, for all g ∈ E′,

⟨f1(z), g

⟩=

1

2πi

∫

|w−z0|=r

⟨f(w), g

⟩

w − zdw =

⟨f(z), g

⟩,

we conclude that f and f1 coincide for |z − z0| < r. Let us finally show that f1 is stronglyholomorphic in z1 with |z1 − z0| < r. In fact, for z near z1,


1

z − z1

(1

w − z− 1

w − z1

)− 1

(w − z1)2

becomes arbitrarily small uniformly for w with |w − z0| = r and hence

f ′1(z1) = limz→z1

f1(z)− f1(z1)

z − z1=

1

2πi

∫

|w−z0|=r

f(w)

(w − z1)2dw

Note that

p

(∫

Γ

h(w) dw

)≤

∫

Γ

p(h(w)

)dw

for a compact C1 curve Γ, a continuous function h : Γ → E, and a continuous seminorm pon E.

By iterating the reasonings of the above proof, we immediately obtain the following (cf.Horváth [7], Prop. (1.2.2), p. 62, and Prop. (1.2.7), p. 64).

Proposition 1.5.3. If U ⊂ C is open and E is a separated quasi-complete locally convex

space and f : U −→ E is holomorphic, then f is also C∞ and Cauchy’s integral formula

holds: If z0 ∈ U andz ∈ C; |z− z0| ≤ r

⊂ U, then ∀n ∈ N0, ∀z1 with |z1 − z0| < r :

f (n)(z1) =n!

2πi

∫

|w−z0|=r

f(w)

(w − z1)n+1dw.

The following lemma is a slight generalization of (9.4.4) in Dieudonné [1], p. 212.

Lemma 1.5.4. If U ⊂ C is open and connected and f1 : U −→ E and f2 : U −→ E are

holomorphic functions with values in a separated locally convex space E coinciding on a

subset A ⊂ U with an accumulation point in U, then f1 = f2.

Proof. For each g ∈ E′ the scalar-valued holomorphic functions g f1 and g f2 coincideon A, and hence, by the identity theorem, everywhere in U. By the Hahn-Banach theoremthis implies f1 = f2.

Analogously to the classical case, a holomorphic function can be developed in a Taylorseries (cf. Horváth [7], Teorema (1.2.8), p. 65):

Proposition 1.5.5 Let f : U −→ E be as in Prop. 1.5.3, z0 ∈ U and B the subset of UB =

z ∈ C; |z − z0| ≤ r

⊂ U. Then

∞∑

n=0

f (n)(z0)

n!(z − z0)

n

converges to f(z) in E uniformly on B.

Proof. If r < r1 such thatz ∈ C; |z−z0| ≤ r1

⊂ U, then the geometric series expansion

1

w − z=

1

w − z0

∞∑

n=0

(z − z0w − z0

)nconverges uniformly for |w− z0| = r1 and z ∈ B. This

22 1 Preliminaries

implies the uniform convergence of

∞∑

n=0

f (n)(z0)

n!(z − z0)

n =

=1

2πi

∞∑

n=0

(z − z0)n ·

∫

|w−z0|=r1

f(w)

(w − z0)n+1dw =

1

2πi

∫f(w)

w − zdw = f(z)

on account of Proposition 1.5.3.

Finite sums of holomorphic functions or limits of locally uniformly convergent sequencesof holomorphic functions U −→ E are also holomorphic (cf. Garnir, de Wilde and Schmets[1] Livre III, III, 29, Théorème de K. Weierstraß, p. 498).

Similarly, if u : E −→ F is a continuous linear mapping and f : U −→ E isholomorphic, then u f is also holomorphic (cf. Horváth [7], Cor. (1.1.5), p. 59). Withrespect to bilinear mappings, we have (cf. Grothendieck [1], Rem. 4, p. 40):

Proposition 1.5.6. Let E,F,G be given separated, quasi-complete locally convex spaces,

u : E×F −→ G a bilinear, separately continuous mapping and two holomorphic functions

f : U −→ E, g : U −→ F. Then u(f, g) : U −→ G : z 7−→ u(f(z), g(z)

)is holomorphic

and

u(f, g)′ = u(f ′, g) + u(f, g′).

Proof. Since it is enough to show the weak holomorphy of u(f, g), i.e. the holomorphyof (h u)(f, g), h ∈ G′, we can suppose G = C from the outset. On the other hand,u : F −→

(E′, σ(E′, E)

): y 7−→ u(−, y) is continuous and hence g := ug : U −→ E′ is

holomorphic if E′ carries the weak topology. Hence it suffices to show that u(f, g) = 〈f, g〉is holomorphic, where 〈 , 〉 : E × E′ −→ C is the canonical dual pairing. Finally, forz0 ∈ U,

limz→z0

⟨f(z), g(z)

⟩−⟨f(z0), g(z0)

⟩

z − z0= lim

z→z0

⟨f(z),

g(z)− g(z0)

z − z0

⟩+

+ limz→z0

⟨f(z)− f(z0)

z − z0, g(z0)

⟩=

⟨f(z0), g

′(z0)⟩+⟨f ′(z0), g(z0)

⟩.

This is obvious for the second term, whereas, for the first term, we employ the fact that

the weak convergence ofg(z)− g(z0)

z − z0in E′ implies its uniform convergence on weakly

compact subsets of E by the Mackey-Arens theorem (cf. Robertson and Robertson [1],Ch. III, § 7, Thm. 7, p. 62; Horváth [4], Ch. III, § 5, Thm. 1, p. 205).

As in Grothendieck, [1], Rem. 1, p. 39, and Horváth [7], Cor. (1.2.5), p. 63, let uscharacterize holomorphy by weak holomorphy on dense subsets:

Proposition 1.5.7. Let E be a separated, quasi-complete locally convex space, U ⊂ C an

open set and suppose that f : U −→ E is weakly continuous and g f is holomorphic for

all g in a dense subset M of(E′, σ(E′, E)

). Then f is holomorphic.


Proof. As in the proof of Prop. 1.5.2, we set f1(z) :=1

2πi

∫

|w−z0|=r

f(w) dww − z

for fixed

z0 ∈ U, |z − z0| < r < dist (z0, ∂U). Since f is weakly continuous and(E, σ(E,E′)

)is

quasi-complete (Horváth [4], Ch. 3, § 6, Thm. 1, p. 218), this line integral is well-definedand f1 is weakly and hence also strongly holomorphic. On the other hand, for g ∈ M, weobtain

g(f1(z)

)=

1

2πi

∫

|w−z0|=r

g(f(w)

)

w − zdw = g

(f(z)

)

and thus, by the density of M, f1(z) = f(z) if |z − z0| < r. This implies the holomorphyof f.

As an application we obtain (cf. Horváth [7], Cor. (1.2.6), p. 64):

Proposition 1.5.8. Let E,F be separated, quasi-complete spaces and let i : E −→ F be

continuous and injective. If f : U −→ E is weakly continuous and i f : U −→ F is

holomorphic, then f is holomorphic as well.

Proof. Since ti : F ′ −→ E′ has dense image in(E′, σ(E′, E)

)(cf. Horváth [4], Ch. III,

§ 12, Cor. 2 to Prop. 2, p. 256) and g i f is holomorphic for each g ∈ F ′, we deducefrom Prop. 1.5.7 that f is holomorphic.

Next let us present an example where the assumption that f : U −→ E is weaklycontinuous can be dispensed with:

Proposition 1.5.9. Let U ⊂ C andΩ ⊂ Rn be open and f : U −→ D′(Ω) be holomorphic

such that f(U) ⊂ E ′(Ω). Then, for each connected component U1 of U, there existsK ⊂ Ωcompact such that supp f(z) ⊂ K for all z ∈ U1, and f : U −→ E ′(Ω) is holomorphic.

Proof. Without loss of generality, let us suppose that U is connected. Let Kj ⊂ Ω be a

sequence of compact sets such that⋃

j∈N

Kj = Ω and set Aj :=z ∈ U ; supp f(z) ⊂ Kj

.

Then Aj ⊂ U is closed since Aj = f−1(D′Kj

)and D′

Kjis closed in D′(Ω). Since U

is of second category and U =⋃

j∈N

Aj , there exists an open subset V, ∅ 6= V ⊂ U,

which is contained in some Aj . But then the holomorphic functions U −→ C : z 7−→⟨ϕ, f(z)

⟩vanish on V if ϕ ∈ D(Ω) with suppϕ ⊂ Ω\Kj . Therefore, these functions

vanish identically on U, i.e., ∀z ∈ U : supp f(z) ⊂ Kj . If χ ∈ D(Ω) fulfills χ(x) = 1 forall x in an open neighbourhood of Kj and ψ ∈ E(Ω), then

U −→ C : z 7−→ 〈ψ, f(z)〉 = 〈ψχ, f(z)〉

is holomorphic, since ψχ ∈ D(Ω). This implies the weak holomorphy of U −→ E ′(Ω) :z 7−→ f(z), and hence also the strong holomorphy due to the completeness of E ′(Ω).

If, instead of E ′, we consider the space of tempered distributions S ′ defined by growthproperties, then the assumption of weak continuity of f : U −→ S ′ cannot be omitted:

Example 1.5.10. There exists an entire function f : C −→ D′(R1) such that f(C) ⊂S(R1) and f : C −→ S ′(R1) is not holomorphic.

24 1 Preliminaries

Proof. For the construction of f, let us adapt a reasoning which goes back to Montel [1],

pp. 318-320. If Kj :=z ∈ C; |Im z| ≤ j, Re z ∈

[−j,−1

j

]∪ [0, j]

, then K1 ⊂ K2 ⊂

· · · and C\Kj is connected. By Runge’s approximation theorem (cf. Hörmander [3], Cor.1.3.2, p. 7; Berenstein and Gay [1], Thm. 3.1.1, p. 214) there exist polynomials Pj(z) suchthat ∀z ∈ Kj :

∣∣Pj(z)− Y (Re z)∣∣ ≤ e−2j .

Take ϕ ∈ D((0, 1)

)\ 0, set τjϕ(x) = ϕ(x− j) and define

f(z) :=∞∑

j=1

ej(Pj+1(z)− Pj(z)

)τjϕ ∈ C∞(R1).

Then f(z) vanishes for x ≤ 1 and all derivatives of f(z) with respect to x decrease forx→ ∞ as e−x. Hence, ∀z ∈ C : f(z) ∈ S(R1). Furthermore, for ψ ∈ D(R), we have

⟨ψ, f(z)

⟩=

M∑

j=1

ej(Pj+1(z)− Pj(z)

) ∫

R

ψ(x)ϕ(x− j) dx,

if M ≥ maxx ∈ R; ψ(x) 6= 0

and hence z 7−→

⟨ψ, f(z)

⟩is a polynomial. Thus

C −→ D′(R1) : z 7−→ f(z) is holomorphic.

Finally, if ψ(x) :=∞∑

j=1

e−j ϕ(x− j) , then ψ ∈ S(R1) and

∫

R

ψ(x)f(z)(x) dx =∞∑

j=1

ej(Pj+1(z)− Pj(z)

)·∫ψ(x)ϕ(x− j) dx

=

∞∑

j=1

ej(Pj+1(z)− Pj(z)

)e−j

∫

R

∣∣ϕ(x− j)∣∣2 dx

= ‖ϕ‖22∞∑

j=1

(Pj+1(z)− Pj(z)

)

= ‖ϕ‖22(Y (Re z)− P1(z)

).

Hence the evaluation of f : C −→ S ′(R1) : z 7−→ f(z) on ψ is not holomorphic whichimplies that f : C −→ S ′(R1) is not holomorphic.

Let us next present a theorem on the analytic continuation of vector-valued holomorphicfunctions stated first in the case of distribution-valued functions in Gel’fand and Shilov [1],Ch. I, app. 2, no. 3, p. 149-151, and in the general case in Horváth [5], Thm. 1.2, p. 147,and [7], Teor. (1.3.1), (1.3.2), pp. 68, 70.

Proposition 1.5.11. Let ∅ 6= V ⊂ U be open subsets of C, U connected, E a separated,

quasi-complete locally convex space and f : V −→ E a holomorphic function. Suppose

furthermore that the scalar-valued functions g f have analytic continuations to U for each

g ∈ E′. Then f can also be continued analytically to U.


Proof. (a) Without restriction of generality, we can assume that V is also connected. Letfi : Vi −→ E; i ∈ I

be the set of holomorphic functions such that V ⊂ Vi ⊂ U, Vi is

open and connected, and fi extends f, i.e. fi∣∣V= f. If i, j ∈ I and g ∈ E′, then the scalar

holomorphic functions g fi and g fj coincide on Vi and on Vj , respectively, with theanalytic continuation of g f. Therefore fi and fj must coincide on Vi ∩ Vj (by the Hahn-

Banach theorem) and thus f can be extended analytically to f defined on W :=⋃

i∈IVi.

(b) It remains to show thatW = U. Let us argue by contradiction. IfU\W is not empty, thenwe can take z0 ∈ (U ∩W ) \W. For sufficiently small r, the circle

z ∈ C; |z− z0| < 2r

is contained in U. If z1 ∈W with |z1 − z0| < r, then we can develop f into a Taylor seriesaround z1, i.e.

f(z) =

∞∑

n=0

an(z − z1)n, an ∈ E,

converging for |z− z1| < r′ for some r′ > 0. This implies g f(z) =∞∑

n=0

g(an)(z− z1)n,

if g ∈ E′ and |z − z1| < r′.

But the last series converges for |z − z1| ≤ r because then |z − z0| < 2r. This implies byCauchy’s estimates that

∣∣g(an)∣∣ = 1

n!

∣∣(g f)(n)(z1)∣∣ ≤ Cr−n ,where C = sup|z−z1|≤r

∣∣g f(z)∣∣.

Hence rnann∈N0 is weakly bounded, and, by Mackey’s theorem, also bounded inE. This

implies that≈f (z) :=

∞∑

n=0

an(z − z1)n converges in E for |z − z1| < r (by Weierstraß’

M-test).≈f is analytic for |z − z1| < r (since g

≈f , g ∈ E′, is analytic) and coincides with

f in W ∩z ∈ C; |z − z1| < r

by the same argument as in (a).

Thus f ,≈f have a common extension to W1 := W ∪

z ∈ C; |z − z1| < r

) W since

z0 ∈W1 – which contradicts the definition of W.

Let us now introduce meromorphic vector-valued functions and concepts related tomeromorphy (cf. Horváth [7], (1.4), p. 74-76).

Definition 1.5.12. Let U ⊂ C be open, D ⊂ U a discrete set, and suppose that

f : U\D −→ E is holomorphic with values in the quasicomplete, locally convex, separated

space E.(a) For z0 ∈ D, we say that f has a pole of order m ∈ N in z0, if (z − z0)

mf(z) can be

extended to a holomorphic function g in (U \D)∪ z0 such that g(z0) 6= 0. In this case f

has a “Laurent expansion” around z0, i.e. f(z) =

∞∑

k=−mak(z − z0)

k with a−m = g(z0) ∈

E. We call a−1 and a0 ∈ E the residue and the finite part of f at z0, respectively, and we

denote it by a−1 = Resz=z0f(z) and a0 = Pfz=z0f(z).(b) f is called meromorphic in U if it has a pole of finite order at each z0 ∈ D.

26 1 Preliminaries

For meromorphic functions, an analogon of Prop. 1.5.6 holds:

Proposition 1.5.13. Let E,F,G be separated, quasi-complete, locally convex spaces,

u : E × F −→ G a bilinear separately continuous mapping, and f, g two meromorphic

functions on U with values in E,F, respectively. Then u(f, g) is a meromorphic function on

U (defined in U \D, D discrete).

If, in particular, f, g have simple poles in z0 ∈ U, then u(f, g) has at most a double pole in

z0 with

Resz=z0

u(f(z) , g(z)

)= u

(Resz=z0

f(z) , Pfz=z0

g(z))+ u

(Pfz=z0

f(z) , Resz=z0

g(z))

Pfz=z0

u(f(z) , g(z)

)= u

(Resz=z0

f(z) , Pfz=z0

g′(z))+ u

(Pfz=z0

f(z) , Pfz=z0

g(z))

+ u(Pfz=z0

f ′(z) , Resz=z0

g(z)).

Proof. If z0 ∈ U, f(z) =

∞∑

k=−mak(z − z0)

k and g(z) =

∞∑

j=−lbj(z − z0)

l, m, l ∈ N0,

and ak ∈ E, bj ∈ F, then (z − z0)m+lu

(f(z), g(z)

)is holomorphic in z0 by Prop. 1.5.6.

Hence u(f, g) is meromorphic in U. Furthermore, if we choose N ∈ N sufficiently largeand write

f(z) =

N∑

k=−mak(z − z0)

k + (z − z0)N+1f1(z) and

g(z) =

N∑

j=−lbj(z − z0)

j + (z − z0)N+1g1(z)

with f1, g1 holomorphic at z0, then one obtains by the bilinearity of u the following Laurentseries for u

(f(z), g(z)

)at z0 :

u(f(z), g(z)

)=

∞∑

i=−m−lci(z − z0)

i with ci =∑

k+j=ik≥−m,j≥−l

u(ak, bj) =i+l∑

k=−mu(ak, bi−k).

In particular if m = l = 1, then Resz=z0

u(f(z) , g(z)

)= c−1 = u(a−1, b0) + u(a0, b−1) =

u(Resz=z0

f(z) , Pfz=z0

g(z))+ u

(Pfz=z0

f(z) , Resz=z0

g(z))

and

Pfz=z0

u(f(z) , g(z)

)= c0 = u(a−1, b1) + u(a0, b0) + u(a1, b−1) =

= u(Resz=z0

f(z) , Pfz=z0

g′(z))+ u

(Pfz=z0

f(z) , Pfz=z0

g(z))+ u

(Pfz=z0

f ′(z) , Resz=z0

g(z)).

Remark. Of course, Prop. 1.5.13 can be applied to the scalar multiplication

u : C× E −→ E : (λ, x) 7−→ λ · x

in a separated, quasi-complete, locally convex space E and then yields formulae for theresidues and finite parts of f · g if f : U −→ C, g : U −→ E are meromorphic. These

1.6 Distribution-Valued Holomorphic Functions 27

formulae generalize assertions in Horváth [5], p. 151, [7], (2.2.8), p. 90, and Horváth, Ortnerand Wagner [1], Lemma 1, p. 431.

1.6 Distribution-Valued Holomorphic Functions

Let us specialize now the general theory in 1.5 to the case where the locally convex spaceE is a space of distributions. First let us describe how the support of holomorphic ormeromorphic distribution-valued functions may vary (cf. Horváth [7], (2.1.2), (2.1.6), p. 76,88).

Proposition 1.6.1. Let f be a meromorphic function on U ⊂ C open and connected, with

values in D′(Ω), and suppose that ∅ 6= U1 ⊂ U is open, A ⊂ Ω is closed, f is holomorphic

on U1 and ∀z ∈ U1 : supp f(z) ⊂ A. Then supp f(z) ⊂ A for all z ∈ U where f is defined

and supp ak ⊂ A for all coefficients ak of the Laurent expansion of f at each z0 ∈ U.

Proof. For ϕ ∈ D(Ω) with suppϕ ∩ A = ∅, the function z 7−→ 〈ϕ, f(z)〉 vanishes forz ∈ U1, and hence by Lemma 1.5.4 on U. This implies supp f(z) ⊂ A for z ∈ U where fis holomorphic and also 〈ϕ, ak〉 = 0.

Let us next treat the composition of holomorphic or meromorphic functions with linearmaps (cf. Ortner [6], Thm. 4 (2)-(4), p. 367).

Proposition 1.6.2. If f is a holomorphic or meromorphic function on the open set U ⊂ C

with values in D′(Ω), then the same holds for z 7−→ ∂α(f(z)

)and for z 7−→ g∗

(f(z)

),

where α∈Nn0 and g∗ denotes the pullback with respect to a C∞−submersion g : Ω1 → Ω.Similarly if f has values in S ′(Rn), then the function z 7−→ F

(f(z)

)is also holomorphic

or meromorphic, respectively. Thereby the coefficients of the Laurent expansion of these

composed functions are also built up by linear composition.

In particular, for a pole z0 of f, we have

∂α(Resz=z0

f(z))= Res

z=z0∂αf(z), ∂α

(Pfz=z0

f(z))= Pfz=z0

∂αf(z),

g∗(Resz=z0

f(z))= Res

z=z0g∗f(z), g∗

(Pfz=z0

f(z))= Pfz=z0

g∗f(z),

F(Resz=z0

f(z))= Res

z=z0Ff(z), F

(Pfz=z0

f(z))= Pfz=z0

Ff(z).

Proof. The assertions follow from the fact that uf : U → F is meromorphic if f : U → Eis meromorphic and u : E → F is a linear continuous mapping between the separated,quasi-complete, locally convex spaces E and F. The Laurent series of u f around z0 ∈ Uis then ∞∑

k=−mu(ak)(z − z0)

k if f(z) =

∞∑

k=−mak(z − z0)

k.

Finally, let us apply Props. 1.5.6 and 1.5.13 to the multiplication and convolution ofdistributions.

28 1 Preliminaries

Proposition 1.6.3. If f, g are meromorphic functions on U ⊂ C, U open, with values

in E(Ω), D′(Ω) or OM (Rn), S ′(Rn) or DLp(Rn), D′Lq(Rn), respectively, then the

mapping z 7−→ f(z)·g(z) is again meromorphic with values in D′(Ω), S ′(Rn), D′Lr(Rn),

respectively, where1

r≤ 1

p+

1

q. If f, g have simple poles at z0 ∈ U, then

Resz=z0

f(z)g(z) = Resz=z0

f(z) Pfz=z0

g(z) + Pfz=z0

f(z) Resz=z0

g(z), and

Pfz=z0

f(z)g(z) = Resz=z0

f(z) Pfz=z0

g′(z) + Pfz=z0

f(z) Pfz=z0

g(z) + Pfz=z0

f ′(z) Resz=z0

g(z).

Proof. This follows immediately from Prop. 1.5.13, taking into account the separate continu-ity of the multiplication mappings (ϕ, T ) 7−→ ϕ · T, E(Ω)×D′(Ω) −→ D′(Ω),OM (Rn)× S ′(Rn) −→ S ′(Rn), DLp(Rn)×D′

Lq(Rn) −→ D′Lr(Rn).

Proposition 1.6.4. If f, g are meromorphic functions on U ⊂ C, U open, with values in

O′C(R

n), S ′(Rn) or D′Lp(Rn), D′

Lq (Rn) respectively, where1

p+

1

q≥ 1, then the function

z 7−→ f(z)∗g(z) is again meromorphic with values in S ′(Rn) or D′Lr (Rn), 1+

1

r=

1

p+1

q,

respectively. If f, g have simple poles at z0 ∈ U, then

Resz=z0

(f(z) ∗ g(z)

)=

(Resz=z0

f(z))∗(Pfz=z0

g(z))+(Pfz=z0

f(z))∗(Resz=z0

g(z))

and

Pfz=z0

(f(z) ∗ g(z)

)=

(Resz=z0

f(z))∗(Pfz=z0

g′(z))+(Pfz=z0

f(z))∗(Pfz=z0

g(z))

+(Pfz=z0

f ′(z))∗(Resz=z0

g(z)).

The proof follows from Prop. 1.5.13 in the same way as for Prop. 1.6.3.

Remark. Props. 1.6.3 and 1.6.4 generalize Horváth [7], (2.1.6), p. 79, (2.1.12), p. 83, andOrtner [6], Prop. 2, p. 367, Prop. 4, p. 373.

Whereas the convolution defined on products of distribution spaces like O′C × S ′

or D′Lp × D′

Lq preserves holomorphy, this is not the case for the general definition ofconvolution given in Def. 1.3.2. There an additional assumption is necessary (cf. Ortner[6], Prop. 3, p. 372).

Proposition 1.6.5. If f, g are meromorphic functions with values in D′(Rn) on U ⊂ C

open such that for all z ∈ U \D, D ⊂ U discrete, f(z) and g(z) are convolvable, and the

mappings

U \D −→ D′L1(Rn)

z 7−→ (ϕ ∗ f(z) )g(z)

are weakly continuous for each ϕ ∈ D(Rn), then z 7→ f(z) ∗ g(z) is meromorphic in Uwith values in D′(Rn).

1.6 Distribution-Valued Holomorphic Functions 29

Proof. For ϕ ∈ D(Rn), the function z 7→ ϕ ∗ f(z) arises as the composition of f with thelinear mapping

D′(Rn) −→ E(Rn)S 7−→ ϕ ∗ S,

and hence it is meromorphic with values in E(Rn). Prop. 1.6.3 then implies the meromorphyof z 7→ (ϕ ∗ f(z) )g(z) =: h(z) with values in D′(Rn). Taking into account the hypothesisthat h : U \ D −→ D′

L1(Rn) is weakly continuous we can apply Prop. 1.5.7 to concludethat h is holomorphic. This yields that

〈ϕ, f(z) ∗ g(z)〉 = 〈1, (ϕ ∗ f(z) )g(z)〉

is holomorphic and hence the holomorphy of z 7→ f(z) ∗ g(z) on U \D. By multiplicationwith (z − z0)

m for sufficiently large m ∈ N, we infer that z 7→ (z − z0)mf(z) ∗ g(z) is

holomorphic at z0 ∈ D and thus that f(z) ∗ g(z) is meromorphic there.

Example 1.6.6. The weak continuity assumption in Prop. 1.6.5 cannot be dispensed with.

Indeed, if ϕ,Kj , Pj are as in Ex. 1.5.10 and g(z) :=

∞∑

j=1

(Pj+1(z) − Pj(z)

)τjϕ, then

g(z) ∈ S(R1) for all z ∈ C. Furthermore, C → D′(R1) : z 7→ g(z) is holomorphic,

but 〈1, g(z)〉 =(Y (Re z) − P1(z)

) 1∫

0

ϕ(x) dx is not holomorphic for Re z = 0 and thus

z 7→ 1 ∗ g(z) is not holomorphic.

Analogously to Prop. 1.5.9, if the convolvability of f and g, i.e. (ϕ ∗ f(z) )g(z) ∈D′L1(Rn), is ensured by the condition of compact support, i.e. (ϕ ∗ f(z) )g(z) ∈ E ′(Rn),

then the weak continuity condition is superfluous:

Proposition 1.6.7. If f, g are meromorphic functions with values in D′(Rn) onU ⊂ C open

such that, for all z ∈ U \D, D ⊂ U discrete, (ϕ ∗ f(z) )g(z) ∈ E ′(Rn), or, equivalently,

∀K ⊂ Rn compact : ∀z ∈ U \D : (K − supp f(z)) ∩ supp g(z) is compact,

then z 7→ f(z) ∗ g(z) is meromorphic in U with values in D′(Rn) and the formulae in

Prop. 1.6.4 hold.

Proof. As in the proof of Prop. 1.6.5, z 7→ (ϕ ∗ f(z) )g(z) =: h(z) is meromorphic withvalues in D′(Rn). Since, by assumption, h(z) ∈ E ′(Rn) for z ∈ U \D, Prop. 1.5.9 impliesthat also U \ D → E ′(Rn) : z 7→ h(z) is holomorphic. This yields the holomorphy ofz 7→ 〈ϕ, f(z)∗g(z)〉 = 〈1, h(z)〉 on U \D. The remaining assertions follow as in the proofof Prop. 1.6.5.

Remarks. 1) The hypotheses of the last proposition contain those of Horváth [7], (2.1.10),p. 82, where it is assumed that there exist closed sets A,B ⊂ Rn, such that supp f(z) ⊂ A,supp g(z) ⊂ B for each z ∈ U \ D and (K − A) ∩ B is a compact set for all compactK ⊂ Rn. In fact, this seemingly stronger hypothesis is equivalent to the one in Prop. 1.6.7,

30 1 Preliminaries

since it can be shown that, for a holomorphic function f : U −→ D′(Rn) on a connectedopen set U ⊂ C, there exists z0 ∈ U such that supp f(z0) = ∪z∈U supp f(z).2) Note that the formulae in Prop. 1.6.4 for the residue and the finite part do not hold ingeneral under the assumptions of Prop. 1.6.5. In fact, if e.g.

f(z) =Y (|x| − 2)|x|z

z log |x| and g(z) =n+ z

zf(−n− z),

then Prop. 1.6.5 applies with D = 0 ⊂ U = C, but Resz=0

f(z) =Y (|x| − 2)

log |x| and

Pfz=0

g(z) = Y (|x| − 2)|x|−n are not convolvable. Explicitly, we have

Resz=0

f(z) ∗ g(z) =∫

|ξ|≥2|x−ξ|≥2

|ξ|−n log( |ξ||x−ξ|

)

log |ξ| log |x− ξ| dξ ∈ C(Rnx),

which cannot be split into two parts as in Prop. 1.6.4.

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Index

analytic continuation, viii, 24, 31, 37, 53,71

approximate unit, 7associated homogeneous, 64associated quasihomogeneous, 60

Bessel potentials, 108Lps , 86

Buck topology, 1, 6, 7

Cauchy principal value, 42Cauchy’s integral formula, 21Cauchy’s integral theorem, 114Cauchy-Riemann kernels, 32, 69, 71Cauchy-Riemann polynomial, 42characteristic, 1, 35, 38, 46, 53, 86C∞-characteristic, 86holomorphic, 39meromorphic, 39, 46

composition formula, 99composition law, 96, 98, 106, 120convolution, viii, 7, 10, 12, 13

for characteristics in spaces of Besselpotentials, 108

of distributions, viii, 2, 11, 12, 17of generalized Hilbert kernels, 103of measures, viii, 1, 5with Cauchy-Riemann kernels, 104with generalized heat kernels, 86with homogeneous distributions with

Lp-characteristics, 90with homogeneous distributions with

non-vanishing C∞-characteristics, 86,95

with quasihomogeneous distributions,viii, 85

with ultrahyperbolic kernels, 86, 111

convolution group, viiiof elliptic operators, viiiof hyperbolic operators, viii, 86of quasihyperbolic operators, viii, 114,

116of singular integral operators, viii, 87,

117of the wave operator, 86of ultrahyperbolic operators, viii, 113

convolution kernels, 66convolvability

of distributions, 1, 12, 13of homogeneous distributions, 99of measures, 1, 7, 10of ultrahyperbolic kernels, 86, 112with elliptic kernel of Marcel Riesz, 86,

96with quasihomogeneous distributions, 85with ultrahyperbolic kernels, 108, 112

convolvable, 1, 7, 10, 13, 16, 17, 100as measures, 8, 101distributions, 12vector-valued distributions, 103

decompositioncharacteristic, 31of quasihomogeneous distributions, 31radial part, 31

distribution-valued functions, viianalytic, viielliptic, 68holomorphic, vii, 2, 27, 54, 71meromorphic, vii, 68quasihomogeneous, viii

distributions, viiialmost homogeneous, 61associated homogeneous, 98

132 Index

homogeneous, 1, 60, 98integrable, 11, 15on hypersurfaces, viii, 2, 4pullback, 2, 4quasihomogeneous, viii, 31, 33, 35, 42,

58vector-valued, 103

division problem, 40, 42, 67, 85

elliptic equationinhomogeneous, 99

Euler’s identity, 59for quasihomogeneous distributions, 34

Eulers differential equationfor quasihomogeneous distributions, 31

finite part, 2, 25, 31, 32, 40, 79, 84Fourier transform, viii, 32

kernels of singular operators, 66functions

bounded, continuous, 6compact support, 5continuous, 5homogeneous, 32hypergeometric, 75, 118meromorphic, 25meromorphic distribution-valued, 27quasihomogeneous, 32weakly continuous, 19

fundamental solutions, vii, viii, 66elliptic homogeneous linear partial

differential operators, 66, 67, 98iterated Cauchy-Riemann operator, viii,

71iterated heat operator, 73iterated Laplace operator, 68, 69iterated ultrahyperbolic operator, viii, 32,

79, 83, 84iterated wave operator, viii, 46, 57, 81,

107Laplace operator, viiquasihomogeneous differential operators,

32with support in half-spaces, 81

Gauß’ hypergeometric function, 31

generalized heat kernel, 72, 86generalized Hilbert kernels, 86

Hadamard’s method of descent, 31, 55heat operator, 73Hilbert transform, 95Hilbert-Riesz kernel

higher, 121holomorphic, 20, 22, 23, 83

strongly, 1, 20weakly, 1, 20

holomorphic functions, 1, 19, 20, 22, 24, 27holomorphy, 22

strong, weak, 1homogeneous distributions

Fourier transform, 32hypergeometric function, 32, 49, 118

iterated heat operator, 73iterated wave operator, 31

kernel of Marcel Riesz, 32, 68, 96Kronecker-Leray form, 2, 37, 75

Laurent expansion, vii, 25, 27, 38, 40Laurent series, 48Lebesgue spaces Lp(Γ ), 4

measureintegrable, 1, 5, 6, 11on Γ , 3Radon, 5, 7Riemann-Stieltjes, 7surface measure, 3

meromorphic, vii, 25, 28, 29meromorphic continuation, 81, 83moments, 36, 48

parametrix method, 89polar coordinates, 3

generalized, 35pole, 25, 40, 83polyharmonic equation, 96, 97pullback, 1, 2, 4, 5, 27, 31, 32, 33, 47, 64,

77, 80, 111

quaishyperbolic matrices, 117

Index 133

quasihomogeneous, viii, 2, 27, 32, 36, 37quasihomogeneous distributions, 31, 33, 42,

58, 59, 62, 85convolution, 85finite parts, 32Fourier transform, 32, 62structure theorem, 31

quasihomogeneous partial differentialoperators, 85

quasihomogeneous polynomials, viii, 40,42

ix1 +R(x′), viiix1 + ix2, viiixm11 + · · ·+ xmn

n , viii, 41quasihyperbolic, 73, 114quasihyperbolic polynomials, 86

Radon measures, 5, 7regularization, 88, 90, 108

of measures, 8residue, 2, 25, 31, 32, 35, 40, 46, 50, 53, 57,

64, 68, 79Riesz kernels, 32, 68, 103

Seidenberg-Tarski’s theorem, 114strict topology, 1, 6structure theorem, 58, 61, 90

Taylor series, 1, 21topology b, 11

ultrahyperbolic equation, 112ultrahyperbolic kernels, 32, 74

wave operator, 106weak continuity, 19, 23weak holomorphy, 22weakly continuous, 22weighted D′

Lp-spaces, viii, 81, 87, 89

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