characterization of harmonic functions by the behavior of ... · furthermore, we characterize...

ORIGINAL PAPER

Characterization of harmonic functions by the behaviorof means at a single point

Ricardo Estrada1

Received: 30 September 2019 / Accepted: 20 November 2019 / Published online: 13 January 2020� Springer Nature Switzerland AG 2020

AbstractWe give a characterization of harmonic functions by a mean value type property at a single

point. We show that if u is real analytic in X; a is a fixed point of X; and if for all

homogeneous polynomials p of degree k the one dimensional function

up rð Þ ¼ZS

u aþrxð Þp xð Þ dx;

is a polynomial of degree k at the most in some interval 0� r\gp; then u is harmonic in X:If u is smooth, and gp ¼ g does not depend on p, then we show that u must be harmonic in

the ball of center a and radius g: We also give a result that applies to distributions.

Furthermore, we characterize harmonic functions by flow integrals around a single point.

Keywords Harmonic functions � Harmonic polynomials � Mean value theorems

Mathematics Subject Classification Primary 31B05 � 33C55 � 35B05; Secondary

46F10

1 Introduction

The mean value property of harmonic functions has been called their most remarkable and

useful property [11]. It can be stated as

u að Þ ¼ 1

C

ZS

u aþrxð Þ dx; ð1:1Þ

if u is a harmonic function defined in a region X of Rn; a 2 X; and the closed ball

x� aj j � r is contained in X: We use the notation S for the unit sphere in Rn and C ¼2pn=2=C n=2ð Þ for its area.

This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira.

& Ricardo [email protected]

1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

SN Partial Differential Equations and Applications

SN Partial Differ. Equ. Appl. (2020) 1:2https://doi.org/10.1007/s42985-019-0003-z(0123456789().,-volV)(0123456789().,-volV)

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https://doi.org/10.1007/s42985-019-0003-z

Interestingly, the converse result is also true, that is, if u satisfies (1.1) whenever the closed

ball x� aj j � r is contained inX then u is harmonic inX; a result first proved by Koebe in 1906

[11]. It is not necessary that (1.1) holds for all r, and a lot of research has been conducted by

assuming that the mean value property holds at eacha for a single value ra or whether it holds for

two fixed values r for alla:Several books and articles survey these ideas [2, 10, 11, 17] as well as

mean value properties for other partial differential equations [4, 16, 21].

The aim of this article is to give a characterization of harmonic functions by a mean

value type property at a single point. Indeed, in Sect. 3 we show that if u is real analytic in

X; a is a fixed point of X; and if for all homogeneous polynomials p of degree k the one

dimensional function

up rð Þ ¼ZS

u aþrxð Þp xð Þ dx ; ð1:2Þ

is a polynomial of degree k at the most in some interval around 0, interval that may

depend on p, then u is harmonic in X:This result does not hold if assume that u is smooth, but we show in Sect. 4 that if all the

up are polynomials of degree k at the most in the same interval around 0, then u must be

harmonic in a ball around a:In fact, in Sect. 5 we give a stronger result, that applies to distributions as long as the

interval is the same for all p; we do assume that u satisfies a distributional smoothness

condition, but just at a:We also characterize harmonic functions by flow integrals. We establish in Sect. 6 that

if the polynomial mean flow

vp rð Þ ¼ZS

ou

onaþ rxð Þ p xð Þ dx ; ð1:3Þ

is a polynomial of degree k � 1 for all homogeneous p of degree k, under proper regularity

conditions on u, then u is harmonic in a ball around a:

2 Preliminaries

In this article we employ the word smooth to mean C1: The notation Bg að Þ is employed for the

open ball x� aj j\g: Our notation for spaces of test functions and distributions is the standard

one [6].

We denote as Pk the space of homogeneous polynomials of degree k in n variables.

Sometimes it is useful to consider Pk as Pk Rnð Þ a space of homogeneous polynomial

functions in Rn; while sometimes we consider it as a space of polynomial functions defined

in S; Pk Sð Þ; since the restriction p pjS is an isomorphism of vector spaces; notice

however that the elements of Pk Sð Þ may have polynomial expressions in S that are not

homogeneous.1 There is an inner product in Pk defined in terms of the coefficients as [7]

p; qf g ¼Xaj j¼k

a!aaba ; ð2:1Þ

ifp xð Þ ¼P

aj j¼k aaxa andq xð Þ ¼

Paj j¼k bax

a:Notice that p; qf g actually equals the following

constant function, p; qf g ¼ p rð Þq xð Þ; where r ¼ rið Þni¼1¼ o=oxið Þni¼1 is the gradient.

1 For example the expression x1 þ x32 does not look homogeneous, but corresponds to an element of P3;

namely x1r2 þ x3

2:


2 of 13 SN Partial Differ. Equ. Appl. (2020) 1:2

We denote as Hk the subspace Pk formed by the harmonic homogeneous poly-

nomials of degree k. Acording to the Gauss decomposition [1], the space Pk Sð Þ can

be decomposed as a2q� k

Hq Sð Þ; a direct sum that is actually an orthogonal sum

with respect both to the inner product (2.1) as well as with respect to the inner

product

q; pð Þ ¼ 1

C

ZS

q xð Þp xð Þ dr xð Þ ; ð2:2Þ

of the space L2 Sð Þ: Notice that if f ;/h i denotes the evaluation of a distribution f 2D0 Sð Þ at a test function / 2 D Sð Þ then

f ;/ð Þ ¼ 1

Cf ;/

� �; ð2:3Þ

if both f and / belong to L2 Sð Þ:We denote as P ¼ a

1k¼0

Pk the space of all polynomials in n variables; one endows Pwith the inductive limit topology [19]. The space of formal power series in n variables will

be denoted as C x1; . . .; xn½ �½ �: Usually one considers it as a topological vector space by

endowing it with the topology of simple convergence of the coefficients of the series, and

with this topology the spaces C x1; . . .; xn½ �½ � and P are dual spaces [19]; the duality can be

given by the formula

S; qh i ¼Xa2Nn

a!aaba ; ð2:4Þ

if S ¼P

a2Nn aaxa 2 C x1; . . .; xn½ �½ � and q ¼

Pa2Nn bax

a 2 P; (2.4) being a finite sum.

Since P � C x1; . . .; xn½ �½ � we can actually consider the evaluation p; qh i if both p and q are

polynomials, and clearly,

p; qh i ¼ p; qf g ; ð2:5Þ

in that case.

We shall employ the extended Pizzetti formula [4]

1

C

ZS

Y xð Þ/ exð Þ dr xð Þ�X1m¼0

DmY rð Þ/ 0ð ÞWn;k;m

ekþ2m; ð2:6Þ

as e ! 0; which holds if / 2 D Rnð Þ and Y 2 Hk: Here Wn;0;0 ¼ 1 and

Wn;k;m ¼ 2mm!n nþ 2ð Þ � � � nþ 2k þ 2m� 2ð Þ ; if k þ m[ 0 : ð2:7Þ

This asymptotic formula is never true for all test functions if we replace Y by a polynomial

of Pk n Hk: When Y ¼ 1 it becomes the usual Pizzetti formula [15].

2.1 The RE decomposition

It is clear what the radial part of a formal power series S 2 C x1; . . .; xn½ �½ � is. Let us now

explain the part of the series S that is a radial multiple of a given polynomial p 2 Pk;denoted as qp Sð Þp: Suppose first that S ¼ q 2 P; then we define qp qð Þ by asking qp qð Þp to


SN Partial Differ. Equ. Appl. (2020) 1:2 of 13 2

be the projection of q onto the subspace of P consisting of radial multiples of p, with the

inner product ;f g; so that if qkþ2m 2 Pkþ2m

qp qkþ2mð Þ ¼qkþ2m; r

2mp� �r2mp; r2mpf g r2m ¼ Dmp rð Þqkþ2m

r2mp; r2mpf g r2m: ð2:8Þ

Notice that this yields that q 2 P is free from radial multiples of p if and only if

Dmp rð Þqjx¼0¼ 0; for m 2 N: Furthermore, we obtain the formula

qp qð Þ ¼X1m¼0

Dmp rð Þqjx¼0

r2mp; r2mpf g r2m; ð2:9Þ

that can be applied when q is replaced by a general formal power series S.

Any S 2 C x1; . . .; xn½ �½ � can be writen as

S ¼Xa2Nn

aaxa; ð2:10Þ

but we would like to rewrite this series in a different way, namely by using the Radial

Expansion decomposition, or RE decomposition [4]. We construct formal power series

R0;R1;R2; . . . such that

S ¼X1k¼0

Rk ; ð2:11Þ

as follows: R0 is the radial part of S, and in general Rk; a series that starts with degree k, is

the part of S consisting of the radial multiples of homogeneous polynomials of degree k,

but free of radial multiples of homogeneous polynomials of degree less than k. The series

Rk can be constructed without needing to know R0; . . .;Rk�1; actually, if Bk is an

orthogonal basis for Hk; with either the inner product ;f g or ;ð Þ; then

Rk ¼X

Ykð Þj2Bk

qY

kð Þj

Sð ÞY kð Þj ; ð2:12Þ

where the part of S that is a radial multiple of a harmonic homogeneous polynomial

Y 2 Hk is qY Sð ÞY; and

qY Sð Þ ¼ 1

Y;Yð ÞX1m¼0

DmY rð ÞS��x¼0

Wn;k;mr2m ¼ 1

Y;Yf gX1m¼0

DmY rð ÞS��x¼0

Wnþ2k;0;mr2m: ð2:13Þ

3 The real analytic case

The aim of this article is to employ the ensuing polynomial spherical mean property of

harmonic functions to characterize them.

Theorem 3.1 Let u be a harmonic function defined in a region X � Rn: Then for each

p 2 Pk the one dimensional function

up;a rð Þ ¼ZS

u aþrxð Þp xð Þ dx ; ð3:1Þ



is a polynomial of degree k at the most in the interval ½0; gaÞ if the ball of center a and

radius ga is contained in X:

Proof If u is harmonic in the ball of center a and radius ga then [1, Thm. 1.31] there are

unique harmonic polynomials Ym 2 Hm for m 2 N such that

u xð Þ ¼X1m¼0

Ym x� að Þ ; ð3:2Þ

uniformly on compacts of the ball. If p 2 Pk thenRSYm xð Þp xð Þ dx ¼ 0 if m[ k; so that

up;a rð Þ ¼Xkm¼0

ZS

Ym xð Þp xð Þ dx

� �rm; ð3:3Þ

is a polynomial of degree k at the most. h

Our first characterization of harmonic functions by the polynomial spherical mean property

at a single point is the following.

Theorem 3.2 Let u be a real analytic function defined in a region X � Rn: Let

a 2 X: Suppose that for all polynomials p 2 Pk there exists gp [ 0 such that for

0� r\gp the function up ¼ up;a is a polynomial of degree k at the most. Then u is

harmonic in X:

Proof Let Y 2 Hk: Then (2.6) yields

uY rð Þ ¼ZS

u aþrxð ÞY xð Þ dx�CX1m¼0

DmY rð ÞujaWn;k;m

rkþ2m; ð3:4Þ

as r ! 0þ: But uY is a polynomial of degree k and we conclude that all the terms in this

expansion, except perhaps the first, vanish,

DmY rð Þuja¼ 0 ; m[ 0 ; ð3:5Þ

and uY ¼ CMYrk where MY is the constant

MY ¼ Y rð ÞujaWn;k;0

: ð3:6Þ

Furthermore, the part of S, the Taylor series of u at a; that is a radial multiple of Y is just

the constant MY times Y:

Let us now decompose the Taylor series as S ¼P1

k¼0 Rk according to the RE expansion

(2.11). We obtain that if Bk is an orthogonal basis for Hk then

Rk ¼X

Ykð Þj2Bk

qY

kð Þj

Sð ÞY kð Þj ¼

XY

kð Þj2Bk

MY

kð Þj

Ykð Þj ; ð3:7Þ



is in fact a very special formal power series: Rk is a harmonic polynomial of degree k,

Rk 2 Hk: We observe next that the Taylor series of u at a converges to u in a neighborhood

V of a; and consequently

u xð Þ ¼X1k¼0

Rk xð Þ ; x 2 V ; ð3:8Þ

is a uniformly convergent series of harmonic functions in V. Hence u is harmonic in V, and

because it is real analytic in X; u is harmonic in X: h

We remark that the conclusion of this theorem remains true if we just assume that for each

polynomial p of degree k there is a polynomial of one variable qp of degree k at the most

such that for all a[ k;

up rð Þ ¼ qp rð Þ þ o rað Þ ; as r ! 0þ: ð3:9Þ

Another equivalent way to express this is by asking the vanishing of the Hadamard finite

part limit [6, Section 2.4]

F:p: limr!0þ

up rð Þra

¼ 0 ; ð3:10Þ

for all polynomials of p degree k and all a[ k:

The same exact argument of the proof of the Theorem 3.2 gives the ensuing charac-

terization of solutions of the equation Dmu ¼ 0:

Theorem 3.3 Let u be a real analytic function defined in a region X � Rn: Let a 2 X:Suppose that for all polynomials p of degree k the one dimensional function up defined by

(3.1) is a polynomial of degree k þ 2m� 2 at the most for 0� r\gp: Then u satisfies the

equation

Dmu ¼ 0 ; ð3:11Þ

in X: Conversely, if u satisfies (3.11) in X then at each a 2 X the function up ¼ up;a is a

polynomial of degree k þ 2m� 2 at the most in the interval ½0; gaÞ if the ball of center aand radius ga is contained in X:

Interestingly, Theorem 3.3 holds even if m ¼ 0: We also have the following characteri-

zation of solutions of Dmu ¼ 0:

Theorem 3.4 Let u be a real analytic function defined in a region X � Rn: Let a 2 X:Suppose that for all polynomials p of degree k

F:p: limr!0þ

up rð Þra

¼ 0 ; for all a[ k þ 2m� 2: ð3:12Þ

Then u satisfies the Eq. (3.11) in X:



4 The smooth case

We shall now consider a corresponding characterization of harmonic functions, and of

solutions of the equation Dmu ¼ 0; by the polynomial mean values at a single point in the

case u is smooth in a ball around that point.

Theorem 4.1 Let u be a smooth function in the ball Bg að Þ: Suppose that for all polyno-

mials p 2 Pk the one dimensional function up rð Þ ¼RSu aþrxð Þp xð Þ dx; is a polynomial

of degree k þ 2m� 2 at the most for 0� r\g: Then u is a solution of Dmu ¼ 0 in Bg að Þ:

Proof In order to better illustrate the ideas, we shall only present the proof in the

case m ¼ 1: In other words, we shall show that if up is a polynomial of degree k at

the most whenever p is, then u is harmonic. Let Bkf g1k¼0 be a family of orthonormal

basis of the spaces Hk; Bk ¼ Yk;l : l 2 Ik

� �; and let us write u aþrxð Þ as a Fourier-

Laplace series,

u aþrxð Þ ¼X1k¼0

Xl2Ik

wk;l rð ÞYk;l xð Þ ; ð4:1Þ

a series that converges uniformly on compacts of Bg að Þ: As it is well known [9, 18], the

functions wk;l are smooth in the interval ½0; gÞ and actually wk;l rð Þ ¼ ck;lrk þ O rkþ1

for some constant ck;l as r ! 0þ:

On the other hand, wk;l is exactly uYk;l

; so that it is a polynomial of degree k at the

most. Consequently wk;l rð Þ ¼ ck;lrk; and thus u equals the sum of harmonic functions

u xð Þ ¼X1k¼0

Xl2Ik

ck;lYk;l x� að Þ ; ð4:2Þ

uniformly on compacts of the ball Bg að Þ: It follows that u is harmonic in Bg að Þ: h

The statement corresponding to Theorem 3.4 does not hold in the smooth case. In fact,

even though the Theorems 3.2 and 4.1 seem very similar, they are not, and the exact

statement of the Theorem 3.2 does not hold in the smooth case, as we now show.

4.1 A counter example

We now construct a smooth function that is not harmonic in any neighborhood of the

origin, but that satisfies both the hypotheses of the Theorems 3.2 and 3.4.

Let q 2 D Rnð Þ be a function that satisfies

q xð Þ ¼ 1 ; if xj j\1 ; q xð Þ ¼ 2 ; if xj j[ 2 : ð4:3Þ

For k ¼ 1; 2; 3; . . . let Yk 2 Hk be a non zero harmonic homogeneous polynomial of degree

k. We can then choose an increasing sequence of positive kkf g1k¼1 such that the function

u xð Þ ¼X1k¼1

q kkxð ÞYk xð Þ ; ð4:4Þ



that converges for all x; since it is a finite sum for x 6¼ 0 and a sum of zeros if x ¼ 0; is

smooth at the origin. The function u is of course smooth for x 6¼ 0; so that, in fact,

u 2 D Rnð Þ:If p is a polynomial of degree k, then up rð Þ is a polynomial of degree k at the most if

0� r� 1=kk but vanishes for r� 2=kk: Therefore, unless it vanishes, up is never a poly-

nomial in the interval ½0; 3=kk�: If g is fixed, then for k large enough, there are polynomials

p of degree k such that up is not a polynomial of degree k in ½0; g�; therefore Theorem 3.1

yields that u is not harmonic in Bg 0ð Þ:

5 The distributional case

We shall now consider the characterization of certain distributions that satisfy the poly-

nomial mean value property at the center of a ball, namely those that are distributionally

smooth at the center of the ball.

We need to recall some ideas used to study the local behavior of distributions

[6, 14, 20]. Let f 2 D0 Rnð Þ be a distribution defined in the ball B ¼ Bg að Þ: We say that

f has the distributional point value L at x ¼ a in the sense of Łojasiewicz [12, 13] if

lime!0

f aþ exð Þ ¼ L ; ð5:1Þ

in D0 Rnð Þ; that is, if for all test functions / 2 D Rnð Þ

lime!0

f aþ exð Þ;/ xð Þh i ¼ L

ZRn

/ xð Þ dx : ð5:2Þ

We use the notation L ¼ f að Þ Łð Þ . If all distributions raf have distributional point values

at x ¼ a for multiindexes aj j � q; then we have the approximation

f aþ exð Þ ¼Xaj j � q

raf að Þa!

xae aj j þ o eqð Þ ; as e ! 0 ; ð5:3Þ

in D0 Rnð Þ:

Definition 5.1 Let f 2 D0 Rnð Þ: We say that f is distributionally smooth at x ¼ a if for all

a 2 Nn the Łojasiewicz point values raf að Þ Łð Þ exist.

There is an equivalent way to understand distributional smoothness at a point.

Lemma 5.2 Let f 2 D0 Rnð Þ: Then f is distributionally smooth at x ¼ a if and only if there

exists a smooth function w 2 E Rnð Þ such that

f aþ exð Þ ¼ w exð Þ þ o e1ð Þ ; ð5:4Þ

as e ! 0 in the space D0 Rnð Þ:

Proof If f is distributionally smooth at x ¼ a then we may employ Borel’s theorem [6,

Thm. 1.5.3] to find a smooth function w 2 E Rnð Þ such that raw 0ð Þ ¼ raf að Þ for all

a 2 Nn: Then w satisfies (5.4). The converse result is clear. h



Next let us consider the polynomial spherical means of distributions. Let f 2 D0 Bg að Þ

:

Let p 2 Pk: Then the mean

up rð Þ ¼ f aþrxð Þ; p xð Þh ix ; ð5:5Þ

is a distribution in the interval ½0; gÞ of the space rkþn�1Deven½0; gÞ 0

given as

up rð Þ; q rð Þ� �

r¼ f aþ xð Þ; xj j1�k�nq xj jð Þ

D Ex; ð5:6Þ

where q 2 rkþn�1Deven½0; gÞ; that is, q rð Þ ¼ rkþn�1s rð Þ for 0� r\g; where s is the

restriction to ½0; gÞ of an even function of Dð�g; gÞ: Formula (5.6) is just the formula for

changing variables to polar coordinates in Rn: Observe also that the notation Rkþn�1 is

used if instead of employing rkþn�1Deven we use rkþn�1Seven [8]; in general if A �g; gð Þ is a

space of test functions over an open interval around the origin, then A½0; gÞ denotes the set

of restrictions of the elements of A �g; gð Þ to ½0; gÞ [3].

There is a version of the extended Pizzetti formula (2.6) for the asymptotic behavior at

the origin of the harmonic polynomial means of distributions that are distributionally

smooth at the center.

Proposition 5.3 Let f be a distribution of the space D0 Rnð Þ that is distributionally smoothat x ¼ a: Let Yk 2 Hk: Then

uY erð Þ ¼ f aþerxð Þ;Y xð Þh i�CX1m¼0

DmY rð Þf jaWn;k;m

rkþ2mekþ2m; ð5:7Þ

as e ! 0 in the space rkþn�1Deven½0;1Þ 0

; that is, if q 2 rkþn�1Deven½0;1Þ then

uY erð Þ; q rð Þh i�CX1m¼0

DmY rð Þf jaWn;k;m

rkþ2m; q rð Þ� �

ekþ2m; ð5:8Þ

as e ! 0: The formula never holds for all test functions q if we replace Y by a polynomial

of Pk n Hk:

Proof Let w be a smooth function in Rn such that f aþ exð Þ ¼ w exð Þ þ o e1ð Þ as in (5.4).

Then if q 2 rkþn�1Deven½0;1Þ; we may apply (2.6) to w to obtain

f aþerxð Þ;Y xð Þh i; q rð Þh i ¼ZS

w erxð ÞY xð Þ dx; q rð Þ� �

þ o e1ð Þ

�CX1m¼0

DmY rð Þwj0Wn;k;m

rkþ2m; q rð Þ� �

ekþ2m;

and (5.8) follows since DmY rð Þwj0¼ DmY rð Þf ja for all m. h

We shall also need the Fourier-Laplace series of distributions [5]. Let f 2 D0 Bg 0ð Þ

: Let

Bkf g1k¼0 be a family of orthonormal basis of the spaces Hk; Bk ¼ Yk;l : l 2 Ik

� �: Then

we can write f xð Þ as a Fourier-Laplace series in the ball Bg 0ð Þ

f rxð Þ ¼X1k¼0

Xl2Ik

fk;l rð ÞYk;l xð Þ; ð5:9Þ



where the fk;l are the distributions f rxð Þ;Y xð Þ� �

of the space rkþn�1Deven½0;1Þ 0

; in the

sense of convergence in D0 Bg 0ð Þ

; that is, for all test functions / 2 D Bg 0ð Þ

we have

f xð Þ;/ xð Þh i ¼X1k¼0

Xl2Ik

fk;l xj jð ÞYk;l x= xj jð Þ;/ xð Þ� �

: ð5:10Þ

We can now give our characterization of harmonic functions in the distributional case.

Theorem 5.4 Let u be a distribution of the space D0 Bg að Þ

that is distributionally smooth

at x ¼ a: Suppose that for all polynomials p 2 Pk the one dimensional distribution

up rð Þ ¼ u aþrxð Þ; p xð Þh ix is a regular distribution given by a polynomial of degree k þ2m� 2 at the most in ½0; gÞ: Then u is a solution of Dmu ¼ 0 in Bg að Þ:

Proof As before we shall consider the case m ¼ 1: Let us write u aþrxð Þ as a Fourier-

Laplace series,P1

k¼0

Pl2Ik

fk;l rð ÞYk;l xð Þ: Then fk;l rð Þ ¼ uYk;l

rð Þ is a regular distribution

given by a polynomial of degree k at the most, while because u is distributionally smooth at

a; the Proposition 5.3 yields that it has a distributional asymptotic behavior of the type

fk;l erð Þ ¼ ck;lrkek þ O ekþ1

as e ! 0 in the space rkþn�1Deven½0; gÞ

0: Consequently

fk;l rð Þ ¼ ck;lrk and thus

u xð Þ ¼X1k¼0

Xl2Ik

ck;lYk;l x� að Þ ; in D0 Bg 0ð Þ

: ð5:11Þ

If we now observe that the space of harmonic functions is a closed subspace of D0 Bg 0ð Þ

;

we conclude that u is harmonic in Bg að Þ: h

We must point out that one needs to ask u to be distributionally smooth at a; as there are

examples of continuous functions that satisfy all the other hypothesis of the Theorem 5.4

but are not harmonic.

Example 5.5 Let Y 2 Hm; where m[ 1: Let q be an integer with 0\q\m and consider

the function defined as

u xð Þ ¼ xj jqY x

xj j

� �; ð5:12Þ

for x 6¼ 0 and u 0ð Þ ¼ 0: Then u is continuous, actually of class Cq�1; andRSu rxð Þp xð Þ dx ¼ 0 if p 2 Hk and k\m; while, if it does not vanish, it is a polynomial of

degree q when k�m: Naturally u is not harmonic.

6 Flow integrals

In this section we give characterizations of harmonic functions by using the behavior of

flow integrals around a single point. In fact, if u is harmonic in a region X; then for all

subregions K with a smooth boundary we clearly have the zero flux property



ZoK

ou

onnð Þ dn ¼ 0 ; ð6:1Þ

where2 ou=on ¼ u;ini denotes the exterior normal derivative of u on the boundary oK: A

converse result was already given in 1906 by Bocher and Koebe, who independently

proved that if ZoB

ou

onnð Þ dn ¼ 0 ; ð6:2Þ

for all balls contained in X then u is harmonic in this region; see [11] and the references in

that survey. Our aim is to characterize harmonic functions by considering the behavior of

the integrals

vp rð Þ ¼ZS

ou

onaþ rxð Þ p xð Þ dx ; ð6:3Þ

around a fixed point a for all polynomials p.

Our main tool is the following asymptotic formula.

Proposition 6.1 Let / be a smooth function in Rn: Then if Y 2 Hk;ZS

/;i exð Þxi Y xð Þ dx�CX1m¼1

2mþ kð ÞDmY rð Þ/j0Wn;k;m

ekþ2m�1; ð6:4Þ

as e ! 0:

Proof Let w xð Þ ¼ xi/;i xð Þ ¼ D/ð Þ xð Þ; where D ¼ xiri is Euler’s differential operator.

Then the asymptotic formula (2.6) yieldsZS

/;i exð Þxi Y xð Þ dx ¼ 1

e

ZS

w exð ÞY xð Þ dx

�CX1m¼0

DmY rð Þwj0Wn;k;m

ekþ2m�1:

If we now observe that for all multi-indices a

raD ¼ aj jra þ Dra; ð6:5Þ

we obtain DmY rð Þwj0¼ 2mþ kð ÞDmY rð Þ/j0; and (6.4) follows. h

The case Y ¼ 1 is already interesting, since the formula

ZS

/;i exð Þxi dx� 2CX1m¼1

mDmY rð Þ/j0Wn;k;m

e2m�1; ð6:6Þ

2 We follow the usual convention that repeated indices are to be summed.



allows us to obtain Saks’ 1932 characterization of harmonic functions [11], namely, if u is

smooth3 and

lime!0

1

e

ZS

u;i aþ exð Þxi dx ¼ 0 ; ð6:7Þ

for all a 2 X; then u is harmonic in X:

Theorem 6.2 Let u be a real analytic function defined in a region X � Rn: Let a 2 X:Suppose that for all polynomials p of degree k the function of one variable vp given by (6.3)

satisfies

F:p: limr!0þ

vp rð Þra

¼ 0 ; for all a[ k � 1: ð6:8Þ

Then u is harmonic in X: In particular, if for all p 2 Pk there exists rp [ 0 such that vp rð Þis a polynomial of degree k � 1 at the most for 0� r� rp; then u is harmonic in X:

Proof We just need to make obvious changes in the proof of the Theorem 3.2. h

We can also give an analog of the Theorem 5.4. Indeed, we just need the distributional

analog of the formula (6.4).

Proposition 6.3 Let f be a distribution of the space D0 Rnð Þ that is distributionally smoothat x ¼ 0: Let Y 2 Hk: Then

rif erxð Þ;xiY xð Þh i�CX1m¼1

2mþ kð ÞDmY rð Þf j0Wn;k;m

ekþ2m�1rkþ2m�1; ð6:9Þ

as e ! 0 in the space rkþnDeven½0;1Þ 0

; that is, if q 2 rkþnDeven½0;1Þ then

rif erxð Þ;xiY xð Þh i; q rð Þh i�CX1m¼1

2mþ kð ÞDmY rð Þf j0Wn;k;m

rkþ2m�1; q rð Þ� �

ekþ2m�1;

ð6:10Þ

as e ! 0:

We immediately obtain the ensuing result.

Theorem 6.4 Let u be a distribution of the space D0 Bg að Þ

that is distributionally smooth

at x ¼ a: Suppose that for all polynomials p 2 Pk the one dimensional distribution vp rð Þ ¼riu aþrxð Þ;xip xð Þh ix is a regular distribution given by a polynomial of degree k � 1 at

the most in ½0; gÞ: Then u is harmonic in Bg að Þ:

3 Using the arguments of [4] we can just assume that u;i are locally integrable.



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characterization of harmonic functions by the behavior of ... · furthermore, we characterize...

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