On Generalized Integral Means and Euler Type Vector Fields

ON GENERALIZED INTEGRAL MEANS

AND EULER TYPE VECTOR FIELDS

CHIA-CHI TUNG

Abstract. Formulas for the Euler vector fields, the Neumann derivatives, andthe Euler as well as Dirichlet product are derived. Extensions to a Riemann

domain of the Gauss operator, the Gauss’ lemma and the related jump for-mulas are given, and the Gauss-Helmholtz representation with ramifications

proved. Examples of elementary solutions to certain modified Laplace opera-tors, applications to pseudospherical harmonics, and characterizations of pseu-

doradial, pseudospherical, nearly holomorphic, and holomorphic functions, areobtained, and constancy criterion for locally Lipschitz, semiharmonic, respec-

tively, weakly holomorphic functions are given.

1. Introduction

In classical analysis the utility of mean-values of functions over spheres and ballsis exemplified in the characterizations of harmonic and subharmonic functions. Alsowell-known is the Bochner-Martinelli multidimensional representation of holomor-phic functions by means of integration over the whole boundary of a piecewisesmooth domain. In view of increased recent interest in non-smooth domains inanalysis and geometry, it seems desirable to consider such mean-value functionsfrom a more general and unified viewpoint. The rationale lies, basically, in the re-mark of V. G. Maz’ya ([8, p. 178]): ”the needs of practical applications as well as theintrinsic logic of the theory itself dictate the consideration of more general curvesand surfaces”. In this respect it is worth noting also the remark of Cialdea ([3,p. 376]) that it is of interest ”to investigate how the classical algorithms of analysis· · · can be generalized in such a way as to have a general range of applicability”.

The Bochner-Martinelli kernel Ka in Cm (in particular, Cauchy kernel in C1) iscompletely universal in that it is independent of the shape of the domain on whichit is applied. Owing to this universal representation and in vew of the decentralizedGauss’ lemma, it is natural to ask: (i) whether the ”Gauss mean-value formula” canbe decentralized and extended to locally Lipschitz functions (of class Cλ) ? and(ii) what functions admit both the (suitably generalized) Gauss mean-value andthe Bochner-Martinelli representation property? In this work such problems areconsidered for a not necessarily smooth domain lying in an ambient semi-Riemanndomain. Particularly of concern here is the problem of how the Gauss’ lemma

Date: December 30, 2009.

1991 Mathematics Subject Classification. Primary: 31C05; Secondary: 32C30, 31B10.Key words and phrases. Euler vector fields, Neumann vector fields, admissible potential, Gauss

mean, Bochner-Martinelli mean, a-pseudospherical, a-pseudoradial.Supports by the ”Globale Methoden in der komplexen Geometrie” Grant of the German re-

search society DFG and the Faculty Improvement Grant of Minnesota State University, Mankato,

are gratefully acknowledged.

1

2 CHIA-CHI TUNG

and the Helmholtz representation can be coalesced on such domains. Towards thisend, and in keeping with physicists’ interest in ”complexifying” partial differentialequations, suitably modified Laplace operators need to be introduced.

Let (X, p) be a semi-Riemann domain (see §2 for notations) and D ⊆ X arelatively compact open subset. For measurable functions f and ψ on D, set

(f, ψ)D

:=1

4m

∫

D

f ψ υmp .

provided the integral exists. Let 4p denote the induced Euclidean Laplace op-erator on D∗. Let a ∈ X and a′ := p(a). To each given continuous functiong : D\p−1(a′) → C, there is associated a modified Laplace operator, Lg, defined by

Lg(Q) := − ddc (Q υm−1p ) +

1

4mg υm

p , (1.1)

acting on twice continuously differentiable functions Q. A solution Ψ ∈ C2(D\p−1(a′))to the equation Lg(Ψ) = 0, satistified almost everywhere in D, is said to be adapted

to g. Define the pull-back of the Dirac distribution δw, where w ∈W\Σ, by

p∗Dδw(ψ) = 〈〈ψ〉〉

p,D(w), ∀ψ ∈ C2

0(D).

Here the right-hand side is the push-forward of ψ under pcD (see [13, §2]). Denotethe characteristic function of D by 1D , and (for convenience) call an elementqa ∈ C2(D\p−1(a′)) a prekernel on D. A prekernel Ψ = Ψa on D is said to bean elementary solution of the operator Lg if it satisfies the equation

− ddc [1D Ψυm−1p ] +

1

4m[1D g υ

mp ] = p∗

Dδa′ , (1.2)

in the following sense:

− (Ψ, 4pρ)D + (g, ρ)D

= 〈〈ρ〉〉p,D

(a′), ∀ρ ∈ C20(D). (1.3)

Every elementary solution Ψ of Lg is necessarily adapted to g. Similarly, to

each constant λ ∈ C, q ∈ C0(D\p−1(a′)) and η ∈ C0(D), there is associated ametaharmonic operator of type (λ, q, η) on D (denoted by Mλ,q,η, where η isomitted if η ≡ 0) given by

Mλ,q,η(Q) := − ddc (Q υm−1p ) +

1

4m((λ− q)Q− η) υm

p . (1.4)

An elementary solution of the operator Mλ,q,η is a prekernel on D satisfying theequation (1.2) with g replaced by (λ− q)Ψ−η; as such it is necessarily a solutionto the equation

− ddc (Ψυm−1p ) +

1

4m(λ− q)Ψυm

p =1

4mη υm

p , a.e. in D. (1.5)

Elementary solutions for the operator Lg may be sought among those prekernelsqa with elementary singularities (§4). Such a prekernel is an elementary solutionfor the operator Lg if and only if it is adapted to g. Similarly, for the metahar-monic operator Mλ,q,η, the existence of elementary solutions can be ascertainedvia metaharmonic prekernels of type (λ, q, η) (Proposition 4.2).

The universal character of the Bochner-Martinelli kernel as well as the decentral-ized nature of the Gauss’ lemma can be construed as the possibility of defining the

GAUSS MEAN AND EULER VECTOR FIELDS 3

kernel intrinsically. More precisely, given a prekernel qa on D, there is associatedthe kernel form

Ka :=i

2π∂qa ∧ υm−1

p . (1.6)

Also of importance are the kernel forms

Σa := −dcqa ∧ υm−1p ,

Ξa :=1

4πdqa ∧ υm−1

p .(1.7)

If qa is real-valued, then the real, respectively, imaginary, part of Ka is givenby Σa, respectively, Ξa. The Bochner-Martinelli kernel Ka in Cm is the kernelform (1.6) associated to the Newtonian-logarithmic prekernel (4.11). For an ad-

missible prekernel the kernel forms (1.6) and (1.7) are intrinsically linked to theradial and spherical derivatives, Rp,a(ψ), respectively, Sp,a(ψ) (§3), of a functionψ ∈ Cλ(D\A) (where A denotes a thin analytic subset of D).

Suppose a weak Stokes domain D ⊆ X with dD 6= ∅ ([12, p. 568]) is given. Gen-eralizing the Newtonian-logarithmic potential, an admissible potential in a neighbor-hood of D, denoted by qzz∈X , is introduced in §4. Relative to such a potentialdefine, for each ψ ∈ L2(dD), the (decentralized) Gauss mean

G(ψ, z) :=

∫

dD

ψ dc(−qz) ∧ υm−1p , z ∈ X. (1.8)

The mapping G : ψ 7→ Gψ := G(ψ, ) is a compact operator on L2(∂D) (Theorem5.1). Define the direct push-forward of ψ at a by

bψcp,D

(a′) := (〈〈ψ〉〉p,D

+1

2〈〈ψ〉〉

p,∂D) (a′).

The Gauss’ lemma on a double layer potential with unit density ([2, p. 94]) andthe push-forward formula (1.3) can be unified in a generalized form: (Theorem 5.2)

Assume that ψ ∈ C0(D) ∩ Cλ(D\A), where A ⊂ D is a finite subset. Then for

every a ∈ X,

G(ψ, a) = bψcp,D

(a′) − [ψ, qa]D[0] − (ψ, ga)D. (1.9)

Here ga := 4p qa and [ψ, qa]D[0] the limit of the Dirichlet product [ψ, qa]Dnover

Dn := D\TA(ρn), TA(ρn) being (any choice of) a nested sequence of tubular

neighborhoods of A with ρn → 0 (TA(ρn) being empty if so is A).

If S ⊆ X, set ∂S := p−1(p(∂S)). Assume that a ∈ D is properly contained in

D, that is, a ∈ D\∂D (such a point will denoted by a∈D). If qz is a principalquasi-Newtonian family (§4), then, expressing the Dirichlet product [ψ, qz]D as anintegrated radial derivative of ψ on D, one has the decentralized Gauss mean-value

formula: if ψ ∈ Cλ(D\A), then

G(ψ, z) = 〈〈ψ〉〉p,D

(z′) +

∫

D

1

2mr2m−1z

Rp,z(ψ) υmp , ∀z /∈ ∂D. (1.10)

Especially, if for almost every a ∈ D, ψ is a-radially symmetric in a neighborhoodof a, then

4 CHIA-CHI TUNG

G(ψ, a) = 〈〈ψ〉〉p,D

(a′). (1.11)

In view of this (generalized) mean-value property, it makes sense to consider (with

respect to a real admissible prekernel qa) a class of functions ψ ∈ Cλ(D\A), calleda-pseudospherical, such that

limn→∞

G(ψn, a) = 〈〈ψ〉〉p,D\A (a′), (1.12)

where ψn := ψcD\TA(ρn), for any choice of nested tubular neighborhoods TA(ρn).Observe that if A\p−1(a′) is a finite set, then every a-pseudospherical functionψ ∈ Cλ(D\A) ∩ C0(D) has the mean-value property (1.11). From the formula(1.10) it follows that if ψ ∈ Cλ(D) and for all points a off some thin analyticsubset of D, ψ is a-radially symmetric, then ψ is semiharmonic ([12, p. 563])in D (Theorem 6.2). The a-pseudospherical semiharmonic functions may thus bethought of a generalization of the spherical harmonics in Cm (see Propositions 3.2and 3.1).

The classical Helmholtz representation expresses a smooth function on a domainD, with (reasonably smooth boundary) as a sum of three potentials due to a massdensity distributed over D, a single layer of density distributed over ∂D, and adouble layer of density also distributed over ∂D. By decomposing the gradient fieldinto Euler type vector fields and making use of the invariant formulations of theradial and spherical derivatives ((3.8)-(3.9)), the Neumann derivatives are shownto be intrinsically defined (§3); and these formulations lead up to a generalizationof the Helmholtz representation coalesced with the direct push-forward operation:Let qaa∈X be an admissible potential on D for gaa∈X . Assume that ψ ∈C0(D) ∩ C1,1(D\A) where A ⊂ D is a finite subset. Then for every a ∈ X,

bψcp,D

(a′) = G(ψ, a) +

∫

dD

qa dcψ ∧ υm−1

p

− (qa, 4pψ)D + (ga, ψ)D

(1.13)

(Theorem 5.3). The proof of this formula, which is a refinement of the formula(1.9), rests upon the connection between the Laplace operator and the Neumannderivatives via the kernel form of qa. The jump formulas for the Gauss operatorare derived (Proposition 5.1). In a subsequent work a proof of the solvability of aninterior Dirichlet problem on an analytic covering space for a modified Helmholtzequation shall be given.

If ψ ∈ C0(∂D) and ξ ∈ X\∂D, define the Martinelli-Bochner mean of ψ ondD (relative to p[ξ] and the kernel form Kξ) by

[[ψ]]∂D

(ξ) :=

∫

dD

ψ(ζ)Kξ(ζ).

If qa is a real, basic, admissible prekernel and ψ ∈ Cλ(D\A), then for each nestedsequence TA(ρn),

∫

dD

ψ Ξa −∫

D∩dTA(ρn)

ψ Ξa =1

2m

∫

Dn

h

r2m−1a

Sp,a(ψ) υmp , (1.14)


where Dn := D\TA(ρn). Thus, with respect to a real admissible prekernel qa, anelement ψ ∈ Cλ(D\A) is called a-pseudoradial if

limn→∞

∫

dDn

ψ Ξa = 0, (1.15)

where Dn := D\TA(ρn), for some nested sequence TA(ρn). In case A = ∅ thisdefinition amounts to requiring the equality between the generalized Gauss andMartinelli-Bochner means of ψ on dD:

G(ψ, a) = [[ψ]]∂D

(a). (1.16)

A characterization of a-pseudospherical (respectively, a-pseudoradial) functions isgiven in Proposition 6.1. An element ψ ∈ Cλ(D) is called nearly holomorphic in

D with respect to an admissible potential qa for ga, ga ∈ C0(D\p−1(a′)), if

〈ψ, qa〉D = (ψ,−ga)D, ∀a∈D,where

〈ψ, qa〉D :=1

2πi

∫

D

∂ψ ∧ ∂qa ∧ υm−1p . (1.17)

denotes the Euler product of ψ and qa ([13, p. 1587]). If the potential qa is ba-sic, such functions are precisely those admitting the generalized Bochner-Martinelli

representation:

[[ψ]]∂D

(a) = 〈〈ψ〉〉p,D

(a′), ∀a∈D, (1.18)

(Theorem 6.1). If, for every a∈D, ψ is both a-pseudospherical and a-pseudoradial(with respect to some qa as above), then ψ is said to be of GM (Gauss-Martinelli)

type on D. Similarly, if, for a given a∈D, the said conditions hold, then ψ is saidto be of a-GM type. It is a consequence of the formula (5.3) and the push-forwardrepresentation (4.6) that constant functions are of GM type.

It seems of interest to see to what extent the converse of the last assertion istrue. With respect to a given real quasi-Newtonian prekernel qa, a real-valued

locally Lipschitz function ψ ∈ Cλ(D) is of a-GM type if and only if the Euler

product 〈ψ, qa〉D = 0 (Corollary 6.1). Also, in terms of quasi-Newtonian potentials,a strengthening of [6, Theorems 14.1 and 15.1] (see also [13, Theorem 5.1]) can begiven: Let X be a normal Riemann domain in the rest of this section. (Theorem6.3) If ψ ∈ C1,1(D)∩H(D), then ψ is holomorphic in D if and only if ψ satisfies

the boundary condition

∫

dD

(∂nψ) qa dσ = 0, ∀a ∈ D\∂D,

with respect to some principal quasi-Newtonian potential qz on D. While theconstant functions are probably the sole functions of GM type, so far only someweaker results can be ascertained: If ψ ∈ C1,1(D) is semiharmonic in D, then ψis of GM type with respect to some principal quasi-Newtonian potential on D if andonly if it is a constant (Proposition 6.3-(1)). An element ψ ∈ Cλ(D) is a constant

6 CHIA-CHI TUNG

if and only if, locally at each a ∈ D∗, ψ is both a-radially and a-sphericallysymmetric (Theorem 6.4).

2. Preliminaries

Denote by ‖z‖ the Euclidean norm of z = (z1, · · · , zm) ∈ Cm, where eachcomponent zj = xj + i yj . Let the space Cm be oriented so that the EuclideanKahler form υm := ((i/2π) ∂∂ ‖z‖2)m is positive. In what follows let X, Y denote(reduced) complex spaces of dimension m > 0, and p : Y → Cm a holomorphicmap. Set a′ := p(a), p[a] := p − a′, and ra := ‖p[a]‖ for each a ∈ Y. If U ⊆ Yis an open set, a ∈ U and r > 0, set U[a](r) := z ∈ U | ra(z) < r, andU[a][r] := z ∈ U | ra(z) ≤ r. Denote by B[a′ ](r) (respectively, S[a′ ](r)) the openball (respectively, the (2m− 1)-sphere) in Cm with center a′ and radius r, andomit the subscript if a′ = 0. Let dυ (respectively, dσr) denote the Euclideanvolume element of Cm (respectively, the sphere S(r) ). Set |B(r) | := vol (B(r))and | S | := vol (S(1)).

A complex space X together with a holomorphic map p : X →W, where W isa domain in Cm, is called a semi-Riemann domain (of dimension m > 0) if thereexists a thin analytic subset Σ of W with thin inverse image Σp := p−1(Σ) suchthat the restriction p : X0 := X \Σp →W0 := W \Σ has discrete fibers. If Σ = ∅,then (X, p) is a Riemann domain ([5, p. 135]). Every proper holomorphic map ofa pure m-dimensional complex space into a domain W ⊆ C

m of strict rank mis a semi-Riemann domain over W ([1, p. 117]). Unless otherwise mentioned, letp : X → W be a semi-Riemann domain of dimension m > 0. For each a ∈ D0 :=D∩X0, there exists an open neighborhood Ua(ρ) with closure in D0 such that therestriction pcUa is an analytic covering onto a ball U ′ := B[a′ ](ρ) in Cm, called a

pseudoball (of radius ρ) at a ([12, p. 557]). If a ∈ X0, define d(a) := the supremumof R > 0 for which a pseudoball U[a](R) b D exists. Let X∗ be the largest opensubset of X on which p is locally biholomorphic, and set D∗ := D ∩X∗.

The notions of Ck-differential forms, the exterior differentiation d, the operators∂, ∂ and dc := (1/4πi)(∂ − ∂), are well-defined on a complex space Y despite thepresence of singularities (see [11, Chapter 4]). Denote by Cµ(G) the set of allC-valued functions of class Cµ (when µ = β, locally bounded, and µ = λ, locallyLipschitz, functions) on G, and by Ak

µ(G) the set of C-valued k-forms of class

Cµ ([11, §4]) on G. The sets Cµ(G) and Akµ(G) are similarly defined. The set of

all φ ∈ C1(D) with locally Lipschitz partial derivatives φxj

and φyj, 1 ≤ j ≤ m,

is denoted by C1,1(D) ([12, p. 562]).If G ⊆ Y is open subset, denote by dG the (maximal) boundary manifold of

Greg in Yreg, the manifold of simple points of Y, oriented to the exterior of Greg

([11, p. 218]). If p : G → Cm is a holomorphic map and a ∈ G, set dυ :=p∗dυ, dσ[a],r := (p[a])∗dσr, where dυ (respectively, dσr) denotes the Euclideanvolume element of Cm (respectively, S(r)). The form

υp : = ddc r2a =i

2π∂∂ r2a (2.1)

is nonnegative (see [11, §4]) and independent of a. Denote the real and imaginaryparts of the Bochner-Martinelli form Ka by σa and ξa, respectively. In particular,the Poincare form


σa :=2 dcra

r2m−1a

∧ υm−1p

is d-closed. Let jdG

: dG → Y denote the inclusion mapping and dσ = dσU∩dG

the (Lebesgue) surface measure on U ∩ dG induced by the local patches pU

: U →B[a′](r) on an unramified neighborhood U of a point a ∈ dG.

3. Euler and Neumann type vector fields

Given a continuous mapping ξ = (ξ1, · · · , ξ2m) : D∗ → C2m, there is a naturallyassociated (complex) vector field ∂

ξ, which for notational convenience, shall be

identified with ξ. Observe that the Cauchy-Riemann (respectively, anti-Cauchy-

Riemann) vector fields, ∂∂pk

(respectively, ∂∂pk

), 1 ≤ k ≤ m, are defined in X∗

by the gradient of h = 12 pk (respectively, 1

2 pk). More generally, given a function

h ∈ C1(D), define the Euler type vector fields

Eh :=

m∑

k=1

∂h

∂pk

∂∇pk, resp. Eh :=

m∑

k=1

∂h

∂pk

∂∇pk. (3.1)

Then the gradient vector field of h admits a decomposition

∂∇h = Eh + Eh (3.2)

in D∗. Furthermore, for every a ∈ X, taking h = 12 r

2a in the expressions in (3.1),

the Euler vector fields Ep,a, respectively, Ep,a, is defined ([12, p. 567]). Also ofuse is the decomposition

∂i∇h = i (Eh − Eh) (3.3)

in D∗. Here the vector i∇h is defined as an element in the complexified tangentspace of D∗ at a.

Let ρ = 0 be a local C1 defining equation of dD in an open set U ⊆ X∗,with dρ 6= 0 on dD ∩ U and ν := ∇ρ

‖∇ρ‖giving the unit outward normal to

dD. Associated with dD are the the ∂- and ∂-, Neumann vector fields, defined,respectively, by

∂n :=1

‖∇ρ‖ Eρ, ∂n :=1

‖∇ρ‖ Eρ, (3.4)

in U. It follows from the definitions (3.1) and (3.4) that for all ψ ∈ C1(U),

∂nψ = ∂nψ. (3.5)

(For an alternative definition of the ∂-Neumann derivative and further discussions,see [6, p. 62 and pp. 157-159]). To show that the definition (3.4) is independent ofthe choice of the local defining equation of dD, two expressions for the Neumannderivatives shall be needed:

∂nψ =1

2[∂νψ + i ∂iνψ], (3.6)

which can be shown by straightforward computations, and

∂nψ =1

2[∂νψ − i ∂iνψ], (3.7)

8 CHIA-CHI TUNG

verified by using the relations (3.5) and (3.6). Set cm := (−1)m(m−1)

2 , and

lm := cm(m− 1)!

4πm.

It is proved in [12, Proposition 5.1] that for all ψ ∈ C1(D),

lm ∂νψ dσ = j∗dD

(dcψ ∧ υm−1p ). (3.8)

Similar computation shows that

lm ∂iνψ dσ =1

4πj∗dD

(dψ ∧ υm−1p ). (3.9)

The identities (3.8)-(3.9) and the formulas (3.6)-(3.7) yield an intepretation, andalso the invariance of the definitions, of the ∂-, respectively, ∂-, Neumann derivative:

Lemma 3.1.

(∂nψ) dσ =1

4πi lmj∗dD

(∂ψ ∧ υm−1p ); (3.10)

(∂nψ) dσ =−1

4πi lmj∗dD

(∂ψ ∧ υm−1p ). (3.11)

In the remainder of this section let D ⊂ X be an open subset, A a thin analyticsubset of D and a ∈ D.

Definition 1. Let ψ ∈ Cλ(D\A). Define the a-radial derivative Rp,a(ψ), respec-tively, the a-spherical derivative Sp,a(ψ), by the following equations:

Ep,a(ψ) =ra

2[Rp,a − iSp,a] (ψ),

Ep,a(ψ) =ra

2[Rp,a + iSp,a] (ψ),

(3.12)

at every point of D∗\p−1(a′) ∪A where the left-hand sides make sense.

Remark 1. If ψ ∈ Cλ(D\A), then the expressions in (3.12) and (3.2) (respectively,(3.3)) (with h = ra) imply the first (respectively, the second) relation below:

Rp,a(ψ) = ∂∇ra(ψ),

Sp,a(ψ) = ∂i∇ra

(ψ),(3.13)

at every point of D∗\p−1(a′) ∪ A where the right-hand side makes sense. Thusthe Euler vector fields are expressible (on pseudospheres) in terms of the Neumannderivatives by virtue of the identities (3.6)-(3.7).

Lemma 3.2. For each ψ ∈ Cλ(D\A) and l ∈ Z, set ψa,l := rla ψ. Then the

following relations (i)

Rp,a (ψ) =1

rl+1a

[(Ep,a +Ep,a)ψa,l − l ψa,l], (3.14)

(ii)

Sp,a (ψ) =i

rl+1a

(Ep,a − Ep,a)ψa,l, (3.15)

and (iii) for all k, l ∈ Z,


r−ka 4p (φa,k) = 2k r−l−1

a Rp,a(φa,l) + 4p φ+ k (k + 2m− 2l − 2)φa,−2, (3.16)

hold almost everywhere in D∗.

Proof. Clearly the first two formulas hold for l = 0 by the definition (3.12). Thegeneral case follow then from the Leibnizian law of vector fields. The third formulacan be deduced from the first (making use of [12, (2.11)]):

4p (φa,k) = 4p (rka)φ+ 2 ∂

∇rka(φ) + rk

a 4p φ

= k (k + 2m− 2)φa,k−2 + 2k rk−1a Rp,a(φ) + rk

a 4p φ

= k (k + 2m− 2)φa,k−2 + 2k rk−l−2a [(Ep,a +Ep,a)φa,l − l φa,l] + rk

a 4p φ

= k (k + 2m− 2l− 2)φa,k−2 + 2k rk−l−1a Rp,a(φa,l) + rk

a 4p φ.

Definition 2. Let ψ ∈ Cλ(D\A) and l ∈ Z. Then ψ is called: (1) a (solid)

pseudospherical harmonic (of order l) centered at a (ψ ∈ Ha,l(D)), if ψ is: (i)semiharmonic in D\A; (ii) a-homogeneous of order l, namely, ψ satisfies theEuler’s equation

(Ep,a + Ep,a) (ψ) = l ψ almost everywhere in D∗;

(2) a-radially symmetric in D, if ψ is a-homogeneous of order 0; (3) a-sphericallysymmetric in D, if Ep,a(ψ) = Ep,a(ψ) almost everywhere in D∗.

Example 3.1. Set xj := Re (pj), yj := Im (pj), and zj := (p[a])∗zj, for 1 ≤j ≤ m. If f ∈ C1(R) is real-valued and a∈D, set θj(z) := arctan (

yj−yj(a)xj−xj(a) ), or

θj(z) := arc cot (xj−xj(a)yj−yj(a)

), whichever is defined. It is easily verified that: (i) the

function φ := f(∑m

j=1 cj ‖zj‖2), z ∈ D, the cj’s being arbitrary real constants, sat-

isfies the equation = (Ep,a(φ)) = 0 in D∗; (ii) the function ψ := f(∑m

j=1 cjθj), z ∈D\S, where S is thin analytic in D, satisfies the equation < (Ep,a(ψ)) = 0 almost

everywhere in D∗\S. Therefore it follows that φ is a-spherically symmetric, and

ψ a-radially symmetric, in D.

Remark 2. Let φ, ψ ∈ Cλ(D\A). (1) if φ, respectively, ψ, is a-homogeneousof order s, respectively, k, then so is φψ of order s + k. (2) By the identities(3.14)-(3.15), ψ is: (i) a-radially symmetric in D if and only if the function ψa,k

is a-homogeneous of order k for some (hence every) k ∈ Z; (ii) a-sphericallysymmetric in D if and only if so is ψa,k for some (hence every) k ∈ Z. (3) Ifψ (respectively, ψa,k) is a-homogeneous of order s (respectively, of order s+ k ),then the formula (3.16) yields the relation

r−ka 4p (ψa,k) = 4p ψ + k (k + 2m+ 2s− 2)ψa,−2, (3.17)

almost everywhere in D∗. This formula implies the next lemma.

Lemma 3.3. An element ψ ∈ Cλ(D\A) belongs to Ha,s(D\A) if and only if the

function ψa,k ∈ Ha,s+k(D\A) for k = 2 − 2m− 2s.

10 CHIA-CHI TUNG

Proposition 3.1. An element Ψa ∈ Cλ(D) belongs to Ha,s(D) if and only if the

associated function ψ := Ψa,−s is metaharmonic (in the following sense):

4p ψ +s(s+ 2m− 2)

r2aψ = 0 a.e. in D∗;

consequently the restriction of ψ to each pseudosphere S(a, ρ0) := ∂U[a](ρ0

) ρ0∈

(0, d(a)), is a surface pseudospherical harmonic (namely an eigenfunction of the

induced spherical Laplacian on S(a, ρ0)).

Proof. By the formula (3.17), the expression for the ”spherical Laplacian”,

rs+2a 4p (Ψa,−s) = r2a 4p Ψa − s (s+ 2m− 2)Ψa,

holds almost everywhere in D∗. From this the desired conclusions easily follow.

Example 3.2. Assume that f ∈ O(Cm) and f is homogeneous of order k > 0over R (with respect to the real variables xj, yj , 1 ≤ j ≤ m). Then for any point

a ∈ X, the function ψ := f p[a] belongs to Ha,k(X).

Proposition 3.2. Assume that φ ∈ Ha,j(D) ∩ C1,1(D) and ψ ∈ Ha,k(D) ∩C1,1(D), with j 6= k. Then for every pseudoball U = U[a](ρ), ρ ∈ (0, d(a)), (i)∫

dUψ φ dσ = 0; (ii) (ψa,s, φa,l)U = 0, provided s+ l ≥ 1 − 2m.

Proof. Observe that, by the Green’s first identity, one has

∫

dU

(ψ dcφ − φ dcψ) ∧ υm−1p = 0.

The functions φ and ψ being a-homogeneous, this expression together with theidentity (3.8) imply that

(j − k)

∫

dU

ψ φ

ra

dσ = 0,

which proves the first assertion. Now the second assertion is a consequence of thefirst by virtue of the formula

|B(ρ)| 4m

ρ2m(ψa,s, φa,l)U =

∫ ρ

0

[ψ φcU ]a,t |S| t2m+s+l−1 dt.

Lemma 3.4. Assume that qa ∈ C2(D\p−1(a′)) satisfies the equation

− dc qa ∧ υm−1p = ha

2 dcra

rqa

∧ υm−1p , a.e. in D∗, (3.18)

for some ha ∈ C1(D) and q ∈ Z[0, 2m− 1]. Then, for any given f ∈ Cλ(D), the

following equations hold locally almost everywhere in D∗:

−ha

mrq+1a

Ep,a(f) υmp =

1

2πi∂f ∧ ∂qa ∧ υm−1

p (3.19)

ha

mrq+1a

Ep,a(f) υmp =

1

2πi∂f ∧ ∂qa ∧ υm−1

p ; (3.20)


−ha

2mrqa

Rp,a(f) υmp = df ∧ dcqa ∧ υm−1

p ; (3.21)

ha

2mrqa

Sp,a(f) υmp =

1

4πdf ∧ dqa ∧ υm−1

p . (3.22)

Proof. Let zj be defined as in Example 3.1, and set dz := dz1 ∧ · · · ∧ dzm, dz[j] :=dz1 ∧ · · · ∧ dzj−1 ∧ dzj+1 ∧ · · · ∧ dzm, for 1 ≤ j ≤ m. The equation (3.18) impliesthat locally almost everywhere in D∗,

1

2πi∂f ∧ ∂qa ∧ υm−1

p =ha

2πi

∂r2a

rq+1a

∧ υm−1p ∧ ∂f

=ha

2πi

1

rq+1a

∑

j

zj dzj ∧∑

k

∂f

∂zk

dzk ∧ υm−1p

=ha

2πi

1

rq+1a

( ∑

j

zj

∂f

∂zj

dzj ∧ dzj

)∧ υm−1

p

=ha

2πi

1

rq+1a

(i

2π)m−1(m− 1)!

( ∑

j

zj

∂f

∂zj

)dz1 ∧ dz1 ∧ · · · ∧ dzm ∧ dzm,

from which the formula (3.19) follows. Similarly the relation (3.20) is proved. Byuse of the relations (3.14)-(3.15), the remaining two equations (3.21) and (3.22) canbe easily derived from the formulas (3.19) and (3.20).

For studying properties of Lipschitz functions, integral products of the Dirichletand Euler type are introduced in [13, p. 1587]. Also of use is the following Dirichlet

product of the second kind:

[[f, g]]D

:=1

4π

∫

D

df ∧ dg ∧ υm−1p ,

where f, g ∈ Cλ(D\A), provided the integral exists. In terms of these products theevaluation (and existence proof) of certain singular integrals involving the radial,respectively, spherical derivative, can be attained:

Lemma 3.5. Assume that D is relatively compact in X, and qa ∈ C2(D\p−1(a′))satisfies the equation (3.18) for some ha ∈ C1(D) and q ∈ Z[0, 2m− 1]. Then, for

every f ∈ Cλ(D),

〈f, qa〉D =−1

m

∫

D

ha

rq+1a

Ep,a(f) υmp , (3.23)

[f, qa]D =−1

2m

∫

D

ha

rqa

Rp,a(f) υmp , (3.24)

[[f, qa]]D

=1

2m

∫

D

ha

rqa

Sp,a(f) υmp , (3.25)

where all integrals exist.

12 CHIA-CHI TUNG

Proof. By the expression (3.19), the associated kernel form Ka (given by (1.6)) ofqa satisfies the equation

−∂f ∧ Ka =−ha

mrqa

Ep,a(f) υmp , (3.26)

locally almost everywhere in D∗. The form Ka can be written

Ka(z) = const.

m∑

j=1

(−1)j−1 ha

‖z′ − a′‖q−1

zj

‖z′ − a′‖ dz[j] ∧ dz,

where z′ 6= a′. Hence it follows from [11, Proposition 6.2.8-(1)] that, for all χ ∈A1,µ∩β(D) (that is, χ has measurable and locally bounded coefficients in a local

embedding space), the form χ∧Ka, and similarly χ∧Ka, is locally integrable on D.Especially, the same holds for the form df∧dcqa∧υm−1

p . It follows from the linearity

of the radial differentiation and the equation (3.26) that, for each f ∈ Cλ(D), the

form ha

rq+1a

Ep,a(f) υmp , hence also ha

rqaRp,a(f) υm

p , and harqa

Sp,a(f) υmp , are locally

integrable on D. The desired conclusions are now consequences of the relations(3.26) and (3.21) and (3.22).

4. Elementary solutions to the modified Laplace operator

Concerning the Gauss mean of a continuous function with respect to a (rea-sonably behaved) prekernel, its limiting behavior as the domain decreases is ofprimordial interest. For wider applications the first few results in this section aregiven with more generality than is necessary for the purpose of this paper.

Definition 3. A form Θ ∈ A2m−11 (B\0), where B ⊂ Cm is an open ball, is

called an elementary kernel at 0 ∈ Cm if and only if: (i) Θ(z) =ξ(z)

‖z‖2m−1 , ∀z ∈B\0, for some ξ ∈ A2m−1

1 (B\0, which is locally bounded at 0; (ii) ρ := ‖z‖ ξis of class C1 in B with d ρ = υm; (iv) dΘ is locally integrable at 0.

In this section, let Y denote a general complex space of dimension m > 0 andp : Y → Cm a holomorphic map, G b Y an open subset, and w ∈ Cm such thatthe set G∩ p−1(w) is discrete (or empty). At each c ∈ p−1(w)∩G there exists anopen connected neighborhood ∆ b Y such that the restriction p

∆= p : ∆ → Cm

exhibits ∆ as a pseudoball at c.

Definition 4. An element Λ = Λa ∈ A2m−11 (G\p−1(w)) with a ∈ G is called an

elementary singular form (with elementary singularities along p−1(w) ∩ G), if Λadmits a representation

Λ = αa +∑

l

hla Θl

a, (4.1)

for some αa ∈ A2m−11 (G) and (finitely many) functions hl

a ∈ C1(G) with∑

l hla(z) =

1 for all z ∈ G ∩ p−1(w), where each Θl being an elementary kernel at 0 ∈ Cm

and Θla := (p[a])∗Θl.

Lemma 4.1. Assume that Λa ∈ A2m−11 (G\p−1(w)) is an elementary singular form

and ψ ∈ C0(G). Then: (i) if w 6∈ Cm\p(Spt∂G(ψ)) ([11, §4.2]), the integral


dψe∂G (a) :=

∫

dG

ψ Λa

exists; (ii) denoting by dψec,ε

(a), the above integral for G = Uc(ε) a pseudoball at

c ∈ G0 ∩ p−1(w), one has

limε→0

dψec,ε

(a) = νwp (c)ψ(c). (4.2)

Proof. The existence of the mean-value dψe∂D(a) follows from [11, Lemma 7.1.8].By the equation (4.1), for Uc = Uc(ε),

dψec,ε

(a) =

∫

dUc

ψαa +

∫

dUc

ψ∑

l

hla Θl

a

=

∫

dUc

(ψ − ψ(c))αa + ψ(c)

∫

dUc

αa +∑

l

∫

dUc

ψ hla Θl

a.

Here

∫

dUc

Θla =

∫

dUc

Θlc =

∫

dUc

1

ε2mη =

1

ε2m

∫

Uc

υmp =

∫

dUc

σc.

Then it follows from [12, Proposition 3.1] that

limε→0

dψec,ε

(a) =∑

l

limε→0

∫

dUc

ψ hla Θl

c

=∑

l

νa′

p (c)ψ(c)hla(c) = νa′

p (c)ψ(c).

Proposition 4.1. Assume that G b Y is a weak Stokes domain with dG 6= ∅,ψ ∈ Cλ(G), and w ∈ Cm as above. Assume further that: (i) w /∈ \p(Spt∂G(ψ)),(ii) Λa ∈ A2m−1

1 (G\p−1(w)) is an elementary singular form. Then

〈〈ψ〉〉p,G

(w) = dψe∂G (a) −∫

G

dψ ∧ Λa −∫

G

ψ dΛa; (4.3)

moreover, if G ∩ p−1(w) = ∅, the formula (4.3) remains valid for each a ∈ Y with

a′ /∈ p(G) (the left-hand side being equal to zero).

Proof. Let G ∩ p−1(w) = a1, · · · , as 6= ∅. Choose for each aj a pseudoballUj,ε := Uaj (ε) b G, ε ∈ (0, r∗), such that the Uj,ε’s are pairwise disjoint. Fora = a1 and ε ∈ (0, r∗), integrating the form ψΛa over the boundary manifold ofG\ ∪ Uj,ε, yields, by the Stokes’ theorem ([11, 7.1.3]),

dψe∂G (a) −s∑

j=1

dψeaj ,ε(a) =

∫

G\∪Uj,ε

dψ ∧Λa +

∫

G\∪Uj,ε

ψ dΛa. (4.4)

14 CHIA-CHI TUNG

By Proposition 6.2.8-(1), ibid., for every χ ∈ A1,µ∩β(G), the form χ ∧ Θa isintegrable on G. Let U ⊆ Y be a pseudoball h ∈ C0(U), and u ∈ C1

c (U) be real-valued. To prove that the form h dΘa is integrable on U, it suffices to considerthe case where h is real-valued. Choose a positive function g ∈ C0(U) such thatg + h > 0 in a neighborhood of Spt(u). Writing u = u1 − u2 with continuous,non-negative parts uj, it may be assumed that u ≥ 0. Since the form dΘ islocally integrable the form u ζ dΘa is integrable on U (by [ibid., 5.2.2]). Thusu (ζ + h) dΘa, hence also u h dΘa, is integrable on U. Consequently the formh dΘa is locally integrable in Y. Letting ε → 0 in the formula (4.4) and makinguse of the limit relation (4.2), the formula (4.3) follows. The remaining assertion isa consequence of the Stokes theorem.

To ascertain that certain singular functions give rise to elementary solutions ofa modified Laplace operator (1.1) in a neighborhood of D b X, a refinement ofthe formula (4.3) shall be needed. For this purpose, define, as in [6, p. 2], for eachφ ∈ C1(D), the form

µφ

:=

m∑

k=1

(−1)m+k−1(∂φ

∂pk

) dp[k]

∧ dp (4.5)

at every point of X∗ ∩ D. A prekernel qa ∈ C2(D\p−1(a′)) with a ∈ D is saidto have elementary singularities along p−1(a′)∩D) if so does the associated kernelform Σa = −dcqa ∧ υm−1

p .

Proposition 4.2. Let D b X be a weak Stokes domain with dD 6= ∅ and

ψ ∈ C1,1(D). Assume that w ∈ Cm\p (Spt∂D(ψ)), and qa ∈ C2(D\p−1(w)) has

elementary singularities. If qa is adapted to g ∈ C0(D\p−1(w)), then, setting

qa := cm qa,

− (qa, 4pψ)D

+ (g, ψ)D

= 〈〈ψ〉〉p,D

(w) −∫

dD

(ψ Ka − qa µψ). (4.6)

Proof. By the expressions (4.5) and (1.6), one has

−∂ψ ∧ Ka = ∂ψ ∧ (−i2π

) ∂qa ∧ υm−1p

=(m− 1)!

(2πi)m∂ qa ∧ µ

ψ

=(m− 1)!

(2πi)m(d (qa µψ) − qa

(−1)m

4(4pψ) dp ∧ dp)

=(m− 1)!

(2πi)md (qa µψ) − qa

1

4m(4pψ) υm

p ,

(4.7)

locally in D∗\p−1(w). The formula (4.6) can now be proved in the same way as inProposition 4.1, by making use of the expression (4.7).

Definition 5. An element qa ∈ C2(D\p−1(a′)) with a ∈ X is called an admissible

prekernel on D if: (i) for some ξa ∈ C2(D), the function qa := qa − ξa is real-valued and (ii)


− dc qa ∧ υm−1p = αa + ha

dcr2ar2ma

,∧υm−1p (4.8)

on D\p−1(a′), for some αa ∈ A2m−11 (D), and ha ∈ C1(D) with ha = 1 on

D ∩ p−1(a′);

Remark 3. Observe that each admissible prekernel qa ∈ C2(D\p−1(a′)) with D ∩p−1(a′) 6= ∅ givs rise to an associated kernel Σa possessing elementary singularitiesalong p−1(a′) ∩ D. Consequently, it follows from the formula (4.6) that such aprekernel qa provides an elementary solution to the modified Laplace operator Lg

if and only if it is adapted to g.

A family of admissible prekernels qaa∈X on D is called an admissible potential

on D (for gaa∈X , where ga ∈ C0(D\p−1(a′)), provided that: (i) each qa isadapted to ga; (ii) the (above given) ξa respectively, αa, depends differentiablyto the second, respectively, first, order, on a ∈ X; (iii) the haa∈X has proper

Lipschitzian dependence on a ∈ X in the following sense: at every a ∈ D there isa pseudoball U with the property that for each irreducible branch Uj of U, thereexists gj ∈ Cλ(U ′ × U ′) ∩ C1(U ′ × U ′ \(a′, a′)) with gj(a

′, a′) = 1, such that

hacUj = pUj

∗(gjcU ′ × a′) ∈ Cλ(Uj) ∩ C1(Uj\a). (4.9)

Furthermore, qaa∈X is called: 1) basic, respectively, 2) principal, if, for everya∈D, ξa = αa ≡ 0, respectively, 2) qa is real-valued, basic with ha ≡ 1. Similarterms shall be applied to a specific prekernel qa, provided the above respectivecondition(s) hold for the given function.

Definition 6. Let k ∈ C be a constant, q ∈ C0(D\p−1(a′)) and η ∈ C0(D). Anadmissible potential qaa∈X on D is called: (1) quasi-Newtonian, if each Ψ = qa

is basic and satisfies the Laplace equation L0(Ψ) = 0 almost everywhere in D; (2)metaharmonic of type (λ, q, η), if each Ψ = qa satisfies the equation (1.5).

Example 4.1. Let k ∈ C be a constant. For each (a, y) ∈ X × (X\p−1(a′)),define

γa,m,k

(y) :=

1m−1

exp( ikr2m−2

a

2m−2

)1

r2m−2a

(y), if m > 1,

Y0(kra)(y), if m = 1, k 6= 0,

2 (ln 1ra

)(y), if m = 1, k = 0,

(4.10)

where Y0 denotes the Bessel function of the second kind of order 0. Then qa :=γa,m,k

is an admissible prekernel and, by Proposition (4.2), it gives an elementary

solution of the metaharmonic operator

M(Q) := − ddc (Q υm−1p ) − 1

4mqkQ υm

p ,

where

qk

:=

k2 r4m−6a , if m > 1,

k2 Y0(kra), if m = 1, k 6= 0,

0, if m = 1, k = 0.

16 CHIA-CHI TUNG

If k = 0, the definition (4.10) yields the (principal) Newtonian-logarithmic potential

γa,m

a∈X , where

γa,m

(y) := γa,m,0 (y) =

1

m−11

r2m−2a

(y), if m > 1,

2 (ln 1ra

)(y), if m = 1.(4.11)

Example 4.2. Assume k 6= 0. For each (a, z) ∈ X × (X\p−1(a′)), define the

function

Ha,m,k

(z) :=πkm−1i

2m−1(m− 1)!

1

(ra(z))m−1H

1m−1(kra(z)), z ∈ X\p−1(a′).

Here H1j denotes the Hankel function of the first kind of order j. Then H

a,m,k

(hence also its real part) is a basic metaharmonic prekernel of type (−k2, 1, 0), and

consequently, Ha,m,k

a∈X is a homogeneous metaharmonic potential.

Proof. With s = m− 1, Hs = H1s , and y = kra, one has

dc (1

rsa

Hs(kra)) =[(−s rs

a)Hs + (k

2) rs+1

a Hs−1

] dcr2ar2ma

. (4.12)

From this expression and the properties of the Hankel functions it follows that theform −dcH

a,m,k∧ υm−1

p has singularities as prescribed by the equation (4.8) withαa ≡ 0. Similarly, by taking the d-derivative of both sides of the equation (4.12)it can be shown that

ddc (1

rm−1a

Hm−1(kra)) ∧ υm−1p = (

−k2

2) rm

a Hm−1 dra ∧ σa.

Consequently, the function Ψ = Ha,m,k satisfies the Helmholtz equation with wavenumber k, namely,

− ddc (Ha,m,k υm−1p ) − k2

4mHa,m,k υ

mp = 0 a.e. in D.

Example 4.3. Let λ be a positive constant. By the preceding Example, an elemen-

tary solution of the operator Mλ,1,0(Q) := − ddc (Q υm−1p ) + 1

4mλQ υm

p , is given

by

Ka,m,

√λ(z) :=

λm−1

2

2m−2(m− 1)!

1

(ra(z))m−1Km−1(

√λ ra(z)), z ∈ X\p−1(a′),

where Kj denotes the modified Bessel function of the second kind of order j:

Kj(z′) :=

π

2ij+1H

1j (iz′), z′ ∈ C\0.

Let a ∈ D and qa ∈ C2(D\p−1(a′)) be an admissible prekernel. By the identi-ties (3.10) and (3.7), the associated kernel form Ka (when restricted to the smoothpart of ∂D ) gives rise to the ∂-Neumann derivative

KacdD∩X∗ = lm (∂~n + i ∂~s) qa dσ,


where ~s := iν, ~n := i~s = −ν. This last expression generalizes the decompositionformula of Martinelli [7, (1.3)]. Moreover, this expression implies that the formKacdD is susceptible of an interpretation as an (explicit) sum of a double layerpotential and a tangential derivative of a single layer potential (see Kytmanov [6,p. 8)]).

5. The Gauss Operator

For a function h ∈ C1(D\p−1(a′)) the harmonic residue at c ∈ D is defined by

Resc(h) := limε→0

Resc(h, ε),

provided the limit exists, where

Resc(h, ε) =

∫

dUc(ε)

− dc h ∧ υm−1p ,

([12, (5.12)]). A partial motivation for extending the Gauss’ lemma lies in theproblem of evalulating the total harmonic residues in D (to be considered in a

subsequent work) of a function ψ qa where ψ ∈ Cλ(D) and qa is a prekernel withreasonable singularities along p−1(a′))∩D. In the remainder of this paper, assumethat (X, p) is a Riemann domain and D b X a weak Stokes domain with dD 6= ∅.Theorem 5.1. Assume that qaa∈X is an admissible potential on D. Then: (1)if ψ ∈ L2(∂D), the decentralized Gauss mean (Gψ)(z) := G(ψ, z) (givdn by (1.8))exists; (2) the mapping ψ 7→ Gψ is a compact operator on L2(∂D).

Proof. Let ψ ∈ L2(∂D). By Lemma 4.1, the integral (Gψ)(z) exists if z 6∈ ∂D. To

prove the existence of Gψ at a point a ∈ ∂D, by resorting to a C∞-partitionof unity on ∂D, it is sufficient to consider the case where ψ ∈ L2

0(U ∩ ∂D)for some pseudoball U centered at a. Denote the irreducible branches of U byUj1≤j≤s. The function qa := qa−ξa ∈ C2(D\p−1(a′)) is real-valued and satisfiesthe condition (4.8); in particular, the function ha has the representation (4.9). SetSj := p(∂D ∩Uj) and gj,a′ := gjcU ′ ×a′ ∈ Cλ(U ′)∩C1(U ′\a′). There exists,for each j = 1, · · · , s, a function qj of the form

qj(s, t) = Bj(s, t)1

‖s− t‖2m−2, s, t ∈ U ′, s 6= t, (5.1)

the Bj(s, t) being bounded on Uj ×Uj and of class C1 off the diagonal, such that

j∗Sj(gj,a′2 dcra′

r2m−1a′

∧ υm−1) (y′) = qj(y′, a′) dσ0 y′ ∈ Sj .

Thus the relation

ja∗dD∩Uj

(−dcHa ∧ υm−1p )(y) = ja∗

dD∩Uj(αacUj)(y) + (ja ∗

dD∩UjqacUj)(y) dσ, (5.2)

holds, where q(y, z) := qj(y′, z′) for (y, z) ∈ Uj × Uj , y

′ 6= z′. By the expresion(5.1), qj is a kernel function for Sj of order 2m − 2, hence, by use of polarcoordinates in S with center a′, and modifying the proof of [4, Proposition (3.11)],it can be shown that the form ψ dcqa ∧ υm−1

p is integrable over dD for every

a ∈ ∂D. Therefore the integral

18 CHIA-CHI TUNG

(GUjψ)(z) := −∫

∂D∩Uj

ψ(y) q(y, z) dσ(y), z ∈ ∂D ∩ Uj ,

and consequently

G(ψ, z) =

s∑

j=1

∫

dD∩Uj

ψ dc(−qz) ∧ υm−1p , z ∈ U ∩ ∂D.

exists. Now it can be shown, essentially as in Proposition 12, ibid., that the mappingψ 7→ GUjψ defines a compact operator on L2(∂D ∩ Uj).

A tubular neighborhood, TA(r), of a thin analytic subset A in D is a union ofpseudoballs Ucj = Ucj(r) with centers cj ∈ A and common radius r.

Theorem 5.2. (Generalized Gauss’ lemma) Let qaa∈X be an admissible poten-

tial on D for gaa∈X and A ⊂ D a finite subset. If ψ ∈ C0(D) ∩ Cλ(D\A),then for every a ∈ X,

G(ψ, a) = bψcp,D

(a′) − [ψ, qa]D[0] − (ψ, ga)D. (5.3)

Proof. Let A = c1, · · · , ct. Without loss of generality it may be assumed thatTA(ρn) is a union of disjoint pseudoballs Ucj = Ucj (ρn)\p−1(a′) with ρn → 0. Bythe push-forward formula (4.3) and the relation (3.26), one has, for small ρn > 0,

G(ψ, a) −q∑

j=1

∫

dUcj

ψ dc (−qa) ∧ υm−1p = 〈〈ψ〉〉

p,Dn(a′)

− [ψ, qa]Dn− (ψ,4pqa)Dn

,

(5.4)

where Dn := D\ ∪ T(ρn)(A). It is easy to show that, for every cj ∈ A\p−1(a′),

limn→∞

‖∫

dUcj

(ψ − ψ(cj)) dc qa ∧ υm−1

p ‖ = 0, 1 ≤ j ≤ t. (5.5)

It follows from the equation (4.8) and [11, Proposition 6.2.8-(1)] that the formψ ddcqz ∧ υm−1

p is integrable on D (in fact the integral is a continuous functionof z′ ∈ Cm). Therefore, the equation (1.5) (with λ = 0, q ≡ 0) implies that thefunction ”ψ ga” is integrable on D. Similarly, if ψ is locally Lipschitz on D, thenthe Dirichlet product [ψ, qa]D exists. Thus for each a∈D, by virtue of the formula(4.2), the formula (5.3) follows from the above relation (5.4) by letting n → ∞. If

a /∈ D, the conclusion follows the same way, except that the first term on right-handside (being an empty sum) is equal to zero.

To prove the formula (5.3) for a ∈ ∂D, let p−1(a′) ∩ D = a1, · · · , al, andp−1(a′) ∩ ∂D = b1, · · · , bl′ 6= ∅. Choose pseudoballs Bµ at bµ (respectively Dk

at ak) of radius ε0. For each ε ∈ (0, ε0), set Bµ,ε := Bµ ∩ X[bµ](ε), Dk,ε :=

Dk ∩X[ak](ε), and W = Wε,n := D\(∪l′µ=1Bµ,ε ∪l

k=1Dk,ε∪tj=1Ucj). By the Stokes’

theorem one has


∫

dW

ψ dc qa ∧ υm−1p = [ψ, qa]W + (ψ,4pqa)W , (5.6)

and similarly for the form ξa (in place of qa). At first assume that all bµ lie in

X∗. Set Sε := ∂D\(∂D ∩ ∪l′

j=1Bj,ε). The equations (5.6) and (4.8) that

∫

Sε

ψ dc qa ∧ υm−1p = −

l′∑

µ=1

∫

∂′Bµ,ε

ψ (αa + ha σa) +

l∑

k=1

∫

∂Dk,ε

ψ dc qa ∧ υm−1p

+

t∑

j=1

∫

∂Ucj

ψ dc qa ∧ υm−1p + [ψ, qa]W + (ψ,4pqa)W ,

(5.7)

where ∂′Bµ,ε := ∂Bµ,ε ∩ D. By making use of a local C2-defining equation of∂D at bµ, local analysis shows that the spherical volume of ∂′Bµ,ε tends to thesame limit as that of the half-sphere Zµ,ε := w ∈ ∂Bµ,ε | (w′, ν(bµ)) < 0 (see [9,p. 359]). Thus

∫

∂′Bµ,ε

ψ ha σa =

∫

∂′Bµ,ε

[(ψ − ψ(bµ) ha] σa + ψ(bµ)

∫

∂′Bµ,ε

σa

→ 1

2ψ(bµ), as ε→ 0.

Clearly the integral∫

∂′Bµ,εψαa tends to 0 as ε→ 0. Now consider the case where

the point bµ is a branch point. Let Bµj , j = 1, · · · , sµ, be the branches of Bµ,

each of which contains bµ. The preceding evaluation of the boundary integrals canbe carried out in the same way for the integrals

∫∂′Bµ

j,εψ ∂νqa dσ0 , 1 ≤ j ≤ sµ,

where ∂′Bµj,ε := ∂Bµ

j,ε ∩D. By Theorem 5.1, the form ψ dcqa ∧ υm−1p is integrable

over dD. Consequently, upon letting ε→ 0 and n→ ∞ in the above relation (5.7)and making use of the formulas (4.2) and (3.24), one obtains the formula (5.3).

Remark 4. If, in Theorem 5.2, qzz∈X is a basic metaharmonic potential of type(λ, q, 0) on D, then by the formula (3.24),

G(ψ, a) = bψcp,D

(a′) +

∫

D

ha

2mr2m−1z

Rp,a(ψ) υmp − (λ− q) (ψ, qa)D , ∀a ∈ X.

Following the notations of [10, pp. 2-3], define the interior, respectively, exterior,

boundary trace of ψ ∈ C0(D) at a0 ∈ ∂D by

γint0 ψ (a0) := lim

z∈D→a0

ψ(z), γext0 ψ (a0) := lim

z∈X\ bD→a0

ψ(z), (5.8)

provided the right-hand side limit exists. Similarly, if ψ ∈ C0(∂D) and a0 ∈ dD,define

γint1 ψ (a0) := lim

z∈D→a0

∂νψ (z), γext1 ψ (a0) := lim

z∈X\ bD→a0

∂νψ (z), (5.9)

provided the right-hand side limit exists.

20 CHIA-CHI TUNG

Proposition 5.1. Assume that the restriction p : D → p(D) is an s-sheetedanalytic covering. Let qaa∈X be an admissible potential on D for gaa∈X .

Then for each ψ ∈ C0(D) and for all a0 ∈ ∂D, (i)

γint0 G (ψ, a0) = G(ψ, a0) + s−(a0)ψ(a0); (5.10)

where

s−(a0) :=

s− 1

2〈〈1〉〉

p,∂D(a′0) if D ∩ p−1(a′0) = ∅,

−12 〈〈1〉〉p,∂D (a′0), if D ∩ p−1(a′o) 6= ∅; (5.11)

and (ii)

γext0 G (ψ, a0) = G(ψ, a0) + s+(a0)ψ(a0), (5.12)

where

s+(a0) :=

−1

2〈〈1〉〉

p,∂D(a′0), if D ∩ p−1(a′0) = ∅,

−[s+ 12〈〈1〉〉

p,∂D(a′0)], if D ∩ p−1(a′o) 6= ∅. (5.13)

Proof. By the generalized Gauss’ lemma (5.3), for each z 6∈ ∂D,

G(1, z) = 〈〈1〉〉p,D

(z′) −∫

D

ddcqz ∧ υm−1p ,

where the last integral term is continuous as a function of z′ ([11, Proposition

6.2.8-(1)]). Hence ‖G(1, z)‖ is bounded for all z 6∈ ∂D in a neighborhood of

a0. Also, given z 6∈ ∂D, the formula (5.3) and the identity (3.8) imply that for

ψ := ψ − ψ(a0),

G(ψ, z) = G(ψ, z) + ψ(a0)(〈〈1〉〉

p,D(z′) + (1, gz)D

). (5.14)

Similarly one has

G(ψ, a0) = G(ψ, a0) + ψ(a0)(〈〈1〉〉

p,D+

1

2〈〈1〉〉

p,∂D

)(a′0) + (1, ga0)D

). (5.15)

Thus to prove the relation (5.10), it suffices to consider the following special case:

If ψ ∈ C0(D) vanishes at a0, then

γint0 G (ψ, a0) = G(ψ, a0).

This can be shown by using the expressions (5.1)-(5.2) to estimate the difference

Gψ (z)−Gψ (a0) in two parts, one arising from integrating over that portion of ∂Dthat lies inside small pseudoballs at the bj’s and the other over the rest of ∂D (asin [9, pp. 360-361]). Accordingly the jump formula (5.10) follows from the formulas(5.14) and (5.15). The remaining relation (5.12) can be similarly proved.

Theorem 5.3. (Generalized Gauss-Helmholtz representation) Let qaa∈X be an

admissible potential on D for gaa∈X and A ⊂ D a finite subset. Assume that

ψ ∈ C0(D) ∩ C1,1(D\A). Then for every a ∈ X,


bψcp,D

(a′) = G(ψ, a) +

∫

dD

qa dcψ ∧ υm−1

p

− (qa, 4pψ)D + (ga, ψ)D,

(5.16)

where the second boundary integral on the right is taken in an improper sense.

Proof. By a similar calculation as in proving the formula (4.7), the following relationcan be verified:

−∂ψ ∧ Ka =(m− 1)!(−1)m

(2πi)md (qa µψ) − 1

4mqa (4pψ) υm

p

in D∗. Thus

dψ ∧ dcqa ∧ υm−1p =

−1

2[∂ψ ∧ Ka + ∂ψ ∧ Ka] ∧ υm−1

p

=(m− 1)!

2(2πi)md

(qa µψ + (−1)mqa µψ

)− 1

4mqa (4pψ) υm

p .(5.17)

It is proved in [12, Lemma 5.2] that

(∂nψ) dσU∩dD = 21−mi−m j∗

U∩dD µψ . (5.18)

Therefore by the identity (3.5),

(∂nψ) dσU∩dD = 21−mi−m(−1)m j∗

U∩dD µψ . (5.19)

As in the proof of Theorem 5.2, the formula (5.16) can be deduced from the theformula (4.3), by invoking the expression (5.17) and the identities (3.8)-(3.9) and(3.6)-(3.7), noting that the existence of the integrals (qa, 4pψ)D and (ga, ψ)D

follows the same way as in Proposition 4.1. The ”integral” over dD of the formqa d

cψ ∧ υm−1p is taken to be the limit of the integral of the same over d(D\ ∪

Bj,ε).

6. Pseudoradial and Pseudospherical Functions

A nearly holomorphic function needs not be holomorphic (see Example 6.1). Rel-ative to a basic admissible potential, such functions are, however, precisely thoseadmitting the generalized Bochner-Martinelli representation (1.18). This is a con-sequence of the equations (3.23), (3.26), and the formula (4.3):

Theorem 6.1. (Compare [13, Theorem 5.3]) Assume that qa is a basic admissi-

ble potential for ga, ga ∈ C0(D\p−1(a′)). Then (with respect to qa) a function

ψ ∈ Cλ(D) is nearly holomorphic if and only if ψ admits the representation

[[ψ]]∂D

(a) = 〈〈ψ〉〉p,D

(a′), ∀a∈D.Corollary 6.1. (1) With respect to a basic admissible prekernel qa adapted to

ga ∈ C0(D\p−1(a′)), an element ψ ∈ Cλ(D) is of a-GM type if and only if

〈ψ, qa〉D = [ψ, qa]D = (ψ,−ga)D , ∀a∈D.(2) With respect to a given real quasi-Newtonian prekernel qa, a real-valued ψ ∈Cλ(D) is of a-GM type if and only if 〈ψ, qa〉D = 0.

22 CHIA-CHI TUNG

Unless otherwise mentioned, let A be a thin analytic subset of D, and a aproperly contained point of D.

Proposition 6.1. Assume that qa is a real, admissible prekernel adapted to g ∈C0(D\p−1(a′)) and ψ ∈ Cλ(D\A). (1) ψ is a-pseudospherical if and only if for

some nested sequence (of tubular neighborhoods) TA(ρn),

limn→∞

([ψ, qa]Dn

+ (ψ, g)Dn

)= 0. (6.1)

(2) If qa is basic, then ψ is a-pseudoradial if and only if for some nested sequence

TA(ρn),

limn→∞

[[ψ, qa]]Dn

= 0. (6.2)

Proof. The first assertion is an immediate consequence of the formula (5.3) andthe definition (1.12). To prove the second assertion, suppose at first that ψ isreal-valued. By virtue of the formulas (3.26), (3.25) and (3.12), it can be shown (asin [13, (4.22)]) that, if qa is basic, then

∫

dD

ψΞa = [[ψ, qa]]Dn

+

∫

D∩dTA(ρn)

ψΞa.

Hence ψ is a-pseudoradial if and only if the condition (6.2) holds. Assume nowthat ψ = u + iv is complex-valued. The assumption (6.2) being equivalent to thecondition that the same equation is satisfied by u and v simultaneously, as can beseen by a splitting of the a-spherical derivative into its real and imaginary parts,the conclusion follows from the preceding case of real functions.

Corollary 6.2. If both φ, ψ ∈ Cλ(D\A) are a-radially (respectively, a-spherically)symmetric in D, then φψ is a-pseudoradial with respect to each real, basic, admis-

sible prekernel qa (respectively, a-pseudospherical with respect to each real, quasi-

Newtonian prekernel qa).

Proof. If qa is a real, basic, admissible, respectively, a real quasi-Newtonian, prek-ernel, then by virtue of the equations (3.24) and (3.25), the condition (6.2), respec-tively, (6.1) of Proposition 6.1 reduces to

∫

D[0]

ha

r2m−1a

Sp,a(ψ) υmp = 0,

respectively,

∫

D[0]

ha

r2m−1a

Rp,a(ψ) υmp = 0.

From these the desired conclusions follow.

Example 6.1. Let D ⊆ C2 be a domain containing a = (0, 0), and A :=(z1, z2) ∈ C2| z1 z2 = 0. Define ψ : C2\A→ C by

ψ(z1 , z2) = (z1z2

) − (z2z1

).


Then ψ ∈ Cλ(D\A) and satisfies the equations Ep,a(ψ) = Ep,a(ψ) = 0 in D\A.Hence, by the above Corollary, ψ is of a-GM type with respect to any real, quasi-

Newtonian prekernel qa in D.

Proposition 6.2. Let λ ∈ C, q ∈ C0(D\p−1(a′)), and f ∈ Cλ(∂D). Assume that

ψ ∈ C1,1(D) is a solution of the Neumann problem:

Mλ,q (ψ) = 0 in D∗, ∂νψ = f on dD. (6.3)

Then: (1) the compatibility condition holds:

∫

dD

f dσ =

∫

D

(λ− q)ψ dυ.

(2) With respect to a given metaharmonic prekernel qa of type (λ, q, 0) on D,where a∈D, (i) ψ is a-pseudospherical if and only if

∫

dD

f qa dσ = 0; (6.4)

(ii) ψ is of a-GM type if and only if ψ satisfies the boundary conditions

∫

dD

f qa dσ =

∫

dD

(∂iνψ) qa dσ = 0. (6.5)

Proof. The first assertion is an immediate consequence of the Green’s first identity([12, (5.9)]). The second assertion is a consequence of the representation (5.16) andthe identity (3.8). Under the condition (6.5), by adding up the two integrals in(6.5), the equation (6.4) follows. Hence ψ is a-pseudospherical. Using the identity(5.18), the second equation in (6.5) can be written

∫

dD

qa µψ = 0. (6.6)

Thus by the formula (4.6), ψ is a-pseudoradial, therefore of a-Gauss-Martinellitype.

Conversely, if ψ is of a-GM type on D with respect to some real metaharmonicprekernel qa of type (λ, q, 0), the relation (6.6) holds by Proposition 4.2 and theidentity (5.18). On the other hand, if ψ is a solution to the Neumann problem(6.3), it can be shown as in Proposition 4.2 that, for each a∈D,

〈〈ψ〉〉p,D

(a′) =

∫

dD

(ψ Ka − qa µψ).

From this follows that

∫

dD

qa µψ = 0. (6.7)

The expressions (6.6)-(6.7) and the identities (5.18)-(5.19) imply that the equation(6.5) holds for ψ, thus completing the proof of the Proposition.

24 CHIA-CHI TUNG

Remark 5. For a continuous function φ on the closure of a pseudoball U[a](ρ0) ata ∈ D, the relation

∫

dU[a](ρ)

φ Im(Ka) = 0, ∀ρ ∈ (0, ρ0), (6.8)

holds ([13, (4.4)]). This formula can be rephrased as asserting that φ is centrally

pseudoradial at a (with respect to the Newtonian-logarithmic potential).

If ψ ∈ C1,1(D) is metaharmonic at a ∈ D of type (λ, 0, 0), then for any pseu-doball U[a](ρ0) b D, by the generalized Gauss-Helmholtz representation (5.16),the following relation holds:

νa′

p (a)ψ(a) =

∫

dU[a](ρ)

ψ σa + λ (γa,m

(ρ) (1, ψ)Ua(ρ) − (γa,m

, ψ)Ua(ρ)), (6.9)

for all ρ ∈ (0, ρ0(a)). An element ψ ∈ Cλ(D) is said to be of centralized GM type

at a ∈ D, if the mean-value property (6.9) holds with λ = 0 for all pseudoballsU[a](ρ0) b D. (Observe that ψ is centrally a-pseudoradial).

Corollary 6.3. Assume that ψ ∈ Cλ(D) satisfies one of the following conditions:

(1) ψ is is nearly harmonic ([12, p. 560]); (2) ψ is semiharmonic at a ∈ D; (3)ψ satisfies the equation Ep,a(ψ) = 0 in U∗

a for all pseudoballs U[a] b D. Then ψ

is of centralized GM type at a; especially so is every ψ ∈ O(D∗) ∪ O(D∗).

Proof. Let U = U[a](ρ0) be a pseudoball at a ∈ D. If ψ ∈ Cλ(U) is nearlyharmonic at a, then the mean-value property (6.9) with λ = 0 is a consequenceof (the proof of) Lemma 3.3 and Proposition 3.1 of [12]. If ψ ∈ C1,1(U) ∩ H(D),then the same mean-value property holds by Theorem [13, Theorem 4.2-(2)]. Thesame conclusion follows from [13, Theorem 4.2-(1)], if ψ satisfies the condition (3),in view of the formula (6.8).

Example 6.2. Let U = U[a](ρ0) be a pseudoball and f ∈ Cλ(∂U). A solution

ψ ∈ C1,1(U) of the Neumann problem (6.3) on U (with q = λ = 0) satisfies

necessarily the conditions

∫

dU[a](ρ)

f dσ =

∫

dU[a](ρ)

Sp,a(ψ) dσ = 0, ∀ρ ∈ (0, ρ0).

This is a consequence of Corollary 6.3, Proposition 6.2, and the expressions (4.11)and (3.13).

Theorem 6.2. If a function ψ ∈ Cλ(D) is a-radially symmetric for all points aoff some thin analytic subset of D, then ψ is semiharmonic in D.

Proof. By Corollary 6.2, if ψ ∈ Cλ(D) is locally a-radially symmetric for a givena ∈ D\A, where A being thin analytic in D, then ψ has the centralized Gaussmean-value property (6.9) (with λ = 0) with respect to a pseudoball at a. Henceψ is semiharmonic in D by [12, Theorem 4.2].


Remark 6. As a consequence of the identity [14, (3.8)], a function ψ ∈ C0(Y ) ∩C2(Yreg) is semiharmonic if and only if there exists at each a ∈ Y off a thin analyticsubset, an unramified Riemann covering p

U: U → Ω such that the spherical mean

radial derivative [RpU

,z(ψ)]z,r ≡ 0 at every z ∈ U for sufficiently small r > 0.

Lemma 6.1. Let λ ∈ C, q ∈ C0(D\p−1(a′)), and f ∈ Cλ(∂D). If ψ ∈ C1,1(D)is a solution of the ∂-Neumann problem:

Mλ,q (ψ) = 0 in D∗, ∂nψ = f on dD. (6.10)

then ψ admits the generalized Bochner-Martinelli representation (1.18) for a (given)principal metaharmonic potential qa of type (λ, 0, 0) on D if and only if

∫

dD

f qa dσ = 0, ∀a ∈ D\∂D, (6.11)

Proof. By the formula (4.6) and the identity (5.18), the equation (6.11) is equivalentto the condition that ψ admits the representation (1.18) for each a∈D.

Assume throughout the rest of this section that (X, p) is a normal Riemanndomain. By virtue of Lemma 6.1 and [13, Theorem 5.3], a strengthening of theholomorphicity criterion of [13, Theorem 5.1-(1)] can be given:

Theorem 6.3. If ψ ∈ C1,1(D) is a solution of the ∂-Neumann problem:

−4pψ = 0 in D∗, ∂nψ = f on dD, (6.12)

where f ∈ Cλ(∂D), then ψ ∈ O(D) if and only if ψ satisfies the boundary

condition (6.11) with respect to some (hence all) principal quasi-Newtonian potential

on D.

Proposition 6.3. Let qz be a principal quasi-Newtonian potential on D and

ψ ∈ C1,1(D). (1) If ψ ∈ H(D), then ψ is of GM type with respect to qz if

and only if ψ is a constant. (2) If ψ ∈ O(D∗), then, for each a∈D, ψ is either

a-pseudospherical or a-pseudoradial with respect to qz if and only if ψ is a

constant.

Proof. To prove the first assertion, it suffices to show that, subject to the theboundary conditions

∫

dD

(∂nψ) qa dσ =

∫

dD

(∂nψ) qa dσ = 0, ∀a ∈ D\∂D,

with respect to some principal quasi-Newtonian potential qz on D, ψ is anti-holomorphic in D. By Proposition 6.2-(2) and the identity (3.5), ψ, is of GM typewith respect to qa on D. Thus the expressions (1.11) and (1.16) together yieldthe representation (1.18) for ψ and a∈D. It follows as in the preceding that ψ isholomorphic in D, hence ψ is a constant.

For second assertion, observe that if ψ ∈ Cλ(D), then ψ is a-pseudoradial ifand only if ψ is a-pseudospherical, in view of the representation (1.18). Thereforethe first assertion concludes the proof.

Theorem 6.4. A function ψ ∈ Cλ(D) is a constant if and only if, locally at each

a ∈ D∗, ψ is both a-radially and a-spherically symmetric.

26 CHIA-CHI TUNG

Proof. If a function ψ is, locally at each a ∈ D∗, both a-radially and a-sphericallysymmetric, then by Corollary 6.2, with respect to γ

z,m, ψ is of GM type on U

for every a pseudoball U b D. Therefore Theorem 6.2 and Proposition 6.3-(1)imply that the function ψ is locally a constant in D, hence a constant on D.

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[9] Mikhlin, S. G., Mathematical physics, an advanced course, North-Holland, Amsterdam-London, 1970.

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[11] Tung, C., The first main theorem of value distribution on complex spaces, Memoiredell’Accademia Nazionale dei Lincei, Serie VIII, Vol. XV (1979), Sez.1, 91-263.

[12] Tung, C., Semi-harmonicity, integral means and Euler type vector fields, Advances in AppliedClifford Algebras, 17 (2007), 555-573

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Dept. of Mathematics and Statistics,, Minnesota State University, Mankato, Mankato,

MN 56001

E-mail address : [email protected]

On Generalized Integral Means and Euler Type Vector Fields

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