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

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Abstract

We 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 \(\Omega ,\) \({\mathbf {a}}\) is a fixed point of \(\Omega ,\) and if for all homogeneous polynomials p of degree k the one dimensional function

$$\begin{aligned} \varphi _{p}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+} r\omega \right) p\left( \omega \right) \,\mathrm {d} \omega, \end{aligned}$$

is a polynomial of degree k at the most in some interval \(0\le r<\eta _{p},\) then u is harmonic in \(\Omega .\) If u is smooth, and \(\eta _{p}=\eta \) does not depend on p,  then we show that u must be harmonic in the ball of center \({\mathbf {a}}\) and radius \(\eta .\) We also give a result that applies to distributions. Furthermore, we characterize harmonic functions by flow integrals around a single point.

Introduction

The mean value property of harmonic functions has been called their most remarkable and useful property [11]. It can be stated as

$$\begin{aligned} u\left( {\mathbf {a}}\right) =\frac{1}{C}\int _{\mathbb {S}}u\left( \mathbf {a+}r\omega \right) \,\mathrm {d}\omega, \end{aligned}$$
(1.1)

if u is a harmonic function defined in a region \(\Omega \) of \(\mathbb {R} ^{n},\) \({\mathbf {a}}\in \Omega ,\) and the closed ball \(\left| \mathbf {x-a} \right| \le r\) is contained in \(\Omega .\) We use the notation \(\mathbb {S}\) for the unit sphere in \(\mathbb {R}^{n}\) and \(C=2\pi ^{n/2}/\Gamma \left( n/2\right) \) for its area.

Interestingly, the converse result is also true, that is, if u satisfies (1.1) whenever the closed ball \(\left| \mathbf {x-a}\right| \le r\) is contained in \(\Omega \) then u is harmonic in \(\Omega ,\) 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 each \({\mathbf {a}}\) for a single value \(r_{{\mathbf {a}}}\) or whether it holds for two fixed values r for all \({\mathbf {a}}.\) 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 \(\Omega ,\) \({\mathbf {a}}\) is a fixed point of \(\Omega ,\) and if for all homogeneous polynomials p of degree k the one dimensional function

$$\begin{aligned} \varphi _{p}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+} r\omega \right) p\left( \omega \right) \,\mathrm {d} \omega \,, \end{aligned}$$
(1.2)

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 \(\Omega .\)

This result does not hold if assume that u is smooth, but we show in Sect. 4 that if all the \(\varphi _{p}\) are polynomials of degree k at the most in the same interval around 0,  then u must be harmonic in a ball around \({\mathbf {a}}.\)

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 \({\mathbf {a}}.\)

We also characterize harmonic functions by flow integrals. We establish in Sect. 6 that if the polynomial mean flow

$$\begin{aligned} \chi _{p}\left( r\right) =\int _{\mathbb {S}}\frac{\partial u}{\partial n}\left( {\mathbf {a}}+r\omega \right) \,p\left( \omega \right) \,\mathrm {d}\omega \,, \end{aligned}$$
(1.3)

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 \({\mathbf {a}}.\)

Preliminaries

In this article we employ the word smooth to mean \(C^{\infty }.\) The notation \(B_{\eta }\left( {\mathbf {a}}\right) \) is employed for the open ball \(\left| \mathbf {x-a}\right| <\eta .\) Our notation for spaces of test functions and distributions is the standard one [6].

We denote as \({\mathcal {P}}_{k}\) the space of homogeneous polynomials of degree k in n variables. Sometimes it is useful to consider \({\mathcal {P}}_{k}\) as \({\mathcal {P}}_{k}\left( \mathbb {R}^{n}\right) \) a space of homogeneous polynomial functions in \(\mathbb {R}^{n},\) while sometimes we consider it as a space of polynomial functions defined in \(\mathbb {S},\) \({\mathcal {P}}_{k}\left( \mathbb {S}\right) ,\) since the restriction \(p\rightsquigarrow \left. p\right| _{\mathbb {S}}\) is an isomorphism of vector spaces; notice however that the elements of \({\mathcal {P}}_{k}\left( \mathbb {S}\right) \) may have polynomial expressions in \(\mathbb {S}\) that are not homogeneous.Footnote 1 There is an inner product in \({\mathcal {P}}_{k}\) defined in terms of the coefficients as [7]

$$\begin{aligned} \left\{ p,q\right\} =\sum _{\left| \alpha \right| =k}\alpha !a_{\alpha }{\overline{b}}_{\alpha }\,, \end{aligned}$$
(2.1)

if \(p\left( {\mathbf {x}}\right) =\sum _{\left| \alpha \right| =k}a_{\alpha }{\mathbf {x}}^{\alpha }\) and \(q\left( {\mathbf {x}}\right) =\sum _{\left| \alpha \right| =k} b_{\alpha }{\mathbf {x}}^{\alpha }.\) Notice that \(\left\{ p,q\right\} \) actually equals the following constant function, \(\left\{ p,q\right\} =p\left( \nabla \right) \overline{q\left( {\mathbf {x}}\right) },\) where \(\nabla =\left( \nabla _{i}\right) _{i=1}^{n}=\left( \partial /\partial x_{i}\right) _{i=1}^{n}\) is the gradient.

We denote as \({\mathcal {H}}_{k}\) the subspace \({\mathcal {P}}_{k}\) formed by the harmonic homogeneous polynomials of degree k. Acording to the Gauss decomposition [1], the space \({\mathcal {P}}_{k}\left( \mathbb {S}\right) \) can be decomposed as \(\bigoplus _{2q\le k}{\mathcal {H}} _{q}\left( \mathbb {S}\right) ,\) 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

$$\begin{aligned} \left( q,p\right) =\frac{1}{C}\int _{\mathbb {S}}q\left( \omega \right) {\overline{p}}\left( \omega \right) \,\mathrm {d} \sigma \left( \omega \right) \,, \end{aligned}$$
(2.2)

of the space \(L^{2}\left( \mathbb {S}\right) .\) Notice that if \(\left\langle f,\phi \right\rangle \) denotes the evaluation of a distribution \(f\in {\mathcal {D}}^{\prime }\left( \mathbb {S}\right) \) at a test function \(\phi \in {\mathcal {D}}\left( \mathbb {S}\right) \) then

$$\begin{aligned} \left( f,\phi \right) =\frac{1}{C}\left\langle f,{\overline{\phi }}\right\rangle \,, \end{aligned}$$
(2.3)

if both f and \(\phi \) belong to \(L^{2}\left( \mathbb {S}\right) .\)

We denote as \({\mathcal {P}}=\bigoplus _{k=0}^{\infty }{\mathcal {P}}_{k}\) the space of all polynomials in n variables; one endows \({\mathcal {P}}\) with the inductive limit topology [19]. The space of formal power series in n variables will be denoted as \(\mathbb {C}\left[ \left[ x_{1},\ldots ,x_{n}\right] \right] .\) 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 \(\mathbb {C} \left[ \left[ x_{1},\ldots ,x_{n}\right] \right] \) and \({\mathcal {P}}\) are dual spaces [19]; the duality can be given by the formula

$$\begin{aligned} \left\langle S,q\right\rangle =\sum _{\alpha \in \mathbb {N}^{n} }\alpha !a_{\alpha }b_{\alpha }\,, \end{aligned}$$
(2.4)

if \(S=\sum _{\alpha \in \mathbb {N}^{n}}a_{\alpha } {\mathbf {x}}^{\alpha }\in \mathbb {C}\left[ \left[ x_{1},\ldots ,x_{n}\right] \right] \) and \(q=\sum _{\alpha \in \mathbb {N}^{n} }b_{\alpha }{\mathbf {x}}^{\alpha }\in {\mathcal {P}},\) (2.4) being a finite sum. Since \({\mathcal {P}}\subset \mathbb {C}\left[ \left[ x_{1},\ldots ,x_{n}\right] \right] \) we can actually consider the evaluation \(\left\langle p,q\right\rangle \) if both p and q are polynomials, and clearly,

$$\begin{aligned} \left\langle p,q\right\rangle =\left\{ p,{\overline{q}}\right\} \,, \end{aligned}$$
(2.5)

in that case.

We shall employ the extended Pizzetti formula [4]

$$\begin{aligned} \frac{1}{C}\int _{\mathbb {S}}{\mathsf {Y}}\left( \omega \right) \phi \left( \varepsilon \omega \right) \,\mathrm {d}\sigma \left( \omega \right) \sim \sum _{m=0}^{\infty }\frac{\Delta ^{m}{\mathsf {Y}} \left( \nabla \right) \phi \left( {\mathbf {0}}\right) }{W_{n,k,m} }\varepsilon ^{k+2m}, \end{aligned}$$
(2.6)

as \(\varepsilon \rightarrow 0,\) which holds if \(\phi \in {\mathcal {D}}\left( \mathbb {R}^{n}\right) \) and \({\mathsf {Y}}\in {\mathcal {H}}_{k}.\) Here \(W_{n,0,0}=1\) and

$$\begin{aligned} W_{n,k,m}=2^{m}m!n\left( n+2\right) \cdots \left( n+2k+2m-2\right) \,,\ \ \ \ \ \text {if }\,k+m>0\,. \end{aligned}$$
(2.7)

This asymptotic formula is never true for all test functions if we replace \({\mathsf {Y}}\) by a polynomial of \({\mathcal {P}}_{k}\setminus {\mathcal {H}}_{k}.\) When \({\mathsf {Y}}=1\) it becomes the usual Pizzetti formula [15].

The RE decomposition

It is clear what the radial part of a formal power series \(S\in \mathbb {C} \left[ \left[ x_{1},\ldots ,x_{n}\right] \right] \) is. Let us now explain the part of the series S that is a radial multiple of a given polynomial \(p\in {\mathcal {P}}_{k},\) denoted as \(\rho _{p}\left( S\right) p.\) Suppose first that \(S=q\in {\mathcal {P}};\) then we define \(\rho _{p}\left( q\right) \) by asking \(\rho _{p}\left( q\right) p\) to be the projection of q onto the subspace of \({\mathcal {P}}\) consisting of radial multiples of p,  with the inner product \(\left\{ ,\right\} ,\) so that if \(q_{k+2m} \in {\mathcal {P}}_{k+2m}\)

$$\begin{aligned} \rho _{p}\left( q_{k+2m}\right) =\frac{\left\{ q_{k+2m},r^{2m}p\right\} }{\left\{ r^{2m}p,r^{2m}p\right\} }r^{2m}=\frac{\Delta ^{m}\overline{p}\left( \nabla \right) q_{k+2m}}{\left\{ r^{2m}p,r^{2m}p\right\} }r^{2m}. \end{aligned}$$
(2.8)

Notice that this yields that \(q\in {\mathcal {P}}\) is free from radial multiples of p if and only if \(\left. \Delta ^{m}{\overline{p}}\left( \nabla \right) q\right| _{\mathbf {x=0}}=0,\) for \(m\in \mathbb {N}.\) Furthermore, we obtain the formula

$$\begin{aligned} \rho _{p}\left( q\right) =\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}{\overline{p}}\left( \nabla \right) q\right| _{\mathbf {x=0}} }{\left\{ r^{2m}p,r^{2m}p\right\} }r^{2m}, \end{aligned}$$
(2.9)

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

Any \(S\in \mathbb {C}\left[ \left[ x_{1},\ldots ,x_{n}\right] \right] \) can be writen as

$$\begin{aligned} S=\sum _{\alpha \in \mathbb {N}^{n}}a_{\alpha }{\mathbf {x}} ^{\alpha }, \end{aligned}$$
(2.10)

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 \(R_{0},R_{1},R_{2},\ldots \) such that

$$\begin{aligned} S=\sum _{k=0}^{\infty }R_{k}\,, \end{aligned}$$
(2.11)

as follows: \(R_{0}\) is the radial part of S,  and in general \(R_{k},\) 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 \(R_{k}\) can be constructed without needing to know \(R_{0},\ldots ,R_{k-1};\) actually, if \({\mathfrak {B}}_{k}\) is an orthogonal basis for \({\mathcal {H}}_{k},\) with either the inner product \(\left\{ ,\right\} \) or \(\left(,\right) ,\) then

$$\begin{aligned} R_{k}=\sum _{{\mathsf {Y}}_{j}^{\left( k\right) }\in {\mathfrak {B}}_{k}} \rho _{{\mathsf {Y}}_{j}^{\left( k\right) }}\left( S\right) {\mathsf {Y}} _{j}^{\left( k\right) }\,, \end{aligned}$$
(2.12)

where the part of S that is a radial multiple of a harmonic homogeneous polynomial \({\mathsf {Y}}\in {\mathcal {H}}_{k}\) is \(\rho _{{\mathsf {Y}}}\left( S\right) {\mathsf {Y}},\) and

$$\begin{aligned} \rho _{{\mathsf {Y}}}\left( S\right) =\frac{1}{\left( {\mathsf {Y}},{\mathsf {Y}} \right) }\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}\overline{{\mathsf {Y}} }\left( \nabla \right) S\right| _{\mathbf {x=0}}}{W_{n,k,m} }r^{2m}=\frac{1}{\left\{ {\mathsf {Y}},{\mathsf {Y}}\right\} }\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}\overline{{\mathsf {Y}}}\left( \nabla \right) S\right| _{\mathbf {x=0}}}{W_{n+2k,0,m}}r^{2m}. \end{aligned}$$
(2.13)

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 \(\Omega \subset \mathbb {R}^{n}.\) Then for each \(p\in {\mathcal {P}}_{k}\) the one dimensional function

$$\begin{aligned} \varphi _{p,{\mathbf {a}}}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+}r\omega \right) p\left( \omega \right) \,\mathrm {d}\omega \,, \end{aligned}$$
(3.1)

is a polynomial of degree k at the most in the interval \([0,\eta _{{\mathbf {a}}})\) if the ball of center \({\mathbf {a}}\) and radius \(\eta _{{\mathbf {a}}}\) is contained in \(\Omega .\)

Proof

If u is harmonic in the ball of center \({\mathbf {a}}\) and radius \(\eta _{{\mathbf {a}}}\) then [1, Thm. 1.31] there are unique harmonic polynomials \({\mathsf {Y}}_{m}\in {\mathcal {H}}_{m}\) for \(m\in \mathbb {N}\) such that

$$\begin{aligned} u\left( {\mathbf {x}}\right) =\sum _{m=0}^{\infty }{\mathsf {Y}}_{m}\left( \mathbf {x-a}\right) \,, \end{aligned}$$
(3.2)

uniformly on compacts of the ball. If \(p\in {\mathcal {P}}_{k}\) then \(\int _{\mathbb {S}}{\mathsf {Y}}_{m}\left( \omega \right) p\left( \omega \right) \,\mathrm {d}\omega =0\) if \(m>k,\) so that

$$\begin{aligned} \varphi _{p,{\mathbf {a}}}\left( r\right) =\sum _{m=0}^{k}\left( \int _{\mathbb {S} }{\mathsf {Y}}_{m}\left( \omega \right) p\left( \omega \right) \,\mathrm {d}\omega \right) r^{m}, \end{aligned}$$
(3.3)

is a polynomial of degree k at the most. \(\square \)

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 \(\Omega \subset \mathbb {R}^{n}.\) Let \({\mathbf {a}}\in \Omega .\) Suppose that for all polynomials \(p\in {\mathcal {P}}_{k}\) there exists \(\eta _{p}>0\) such that for \(0\le r<\eta _{p}\) the function \(\varphi _{p}=\varphi _{p,{\mathbf {a}}}\) is a polynomial of degree k at the most. Then u is harmonic in \(\Omega .\)

Proof

Let \({\mathsf {Y}}\in {\mathcal {H}}_{k}.\) Then (2.6) yields

$$\begin{aligned} \varphi _{{\mathsf {Y}}}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+}r\omega \right) {\mathsf {Y}}\left( \omega \right) \,\mathrm {d}\omega \sim C\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) u\right| _{{\mathbf {a}}} }{W_{n,k,m}}r^{k+2m}, \end{aligned}$$
(3.4)

as \(r\rightarrow 0^{+}.\) But \(\varphi _{{\mathsf {Y}}}\) is a polynomial of degree k and we conclude that all the terms in this expansion, except perhaps the first, vanish,

$$\begin{aligned} \left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) u\right| _{{\mathbf {a}}}=0\,,\ \ \ \ m>0\,, \end{aligned}$$
(3.5)

and \(\varphi _{{\mathsf {Y}}}=CM_{{\mathsf {Y}}}r^{k}\) where \(M_{{\mathsf {Y}}}\) is the constant

$$\begin{aligned} M_{{\mathsf {Y}}}=\frac{\left. {\mathsf {Y}}\left( \nabla \right) u\right| _{{\mathbf {a}}}}{W_{n,k,0}}. \end{aligned}$$
(3.6)

Furthermore, the part of S,  the Taylor series of u at \({\mathbf {a}},\) that is a radial multiple of \({\mathsf {Y}}\) is just the constant \(M_{{\mathsf {Y}}}\) times \(\mathsf {Y.}\)

Let us now decompose the Taylor series as \(S=\sum _{k=0}^{\infty }R_{k}\) according to the RE expansion (2.11). We obtain that if \({\mathfrak {B}} _{k}\) is an orthogonal basis for \({\mathcal {H}}_{k}\) then

$$\begin{aligned} R_{k}=\sum _{{\mathsf {Y}}_{j}^{\left( k\right) }\in {\mathfrak {B}}_{k}} \rho _{{\mathsf {Y}}_{j}^{\left( k\right) }}\left( S\right) {\mathsf {Y}} _{j}^{\left( k\right) }=\sum _{{\mathsf {Y}}_{j}^{\left( k\right) } \in {\mathfrak {B}}_{k}}M_{{\mathsf {Y}}_{j}^{\left( k\right) }}{\mathsf {Y}} _{j}^{\left( k\right) }\,, \end{aligned}$$
(3.7)

is in fact a very special formal power series: \(R_{k}\) is a harmonic polynomial of degree k\(R_{k}\in {\mathcal {H}}_{k}.\) We observe next that the Taylor series of u at \({\mathbf {a}}\) converges to u in a neighborhood V of \({\mathbf {a}},\) and consequently

$$\begin{aligned} u\left( {\mathbf {x}}\right) =\sum _{k=0}^{\infty }R_{k}\left( {\mathbf {x}} \right) \,,\text { }{\mathbf {x}}\in V, \end{aligned}$$
(3.8)

is a uniformly convergent series of harmonic functions in V. Hence u is harmonic in V,  and because it is real analytic in \(\Omega ,\) u is harmonic in \(\Omega .\) \(\square \)

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 \(q_{p}\) of degree k at the most such that for all \(\alpha >k,\)

$$\begin{aligned} \varphi _{p}\left( r\right) =q_{p}\left( r\right) +o\left( r^{\alpha }\right) \,,\ \ \ \ \text {as }r\rightarrow 0^{+}. \end{aligned}$$
(3.9)

Another equivalent way to express this is by asking the vanishing of the Hadamard finite part limit [6, Section 2.4]

$$\begin{aligned} \mathrm {F.p.}\lim _{r\rightarrow 0^{+}}\frac{\varphi _{p}\left( r\right) }{r^{\alpha }}=0\,, \end{aligned}$$
(3.10)

for all polynomials of p degree k and all \(\alpha >k.\)

The same exact argument of the proof of the Theorem 3.2 gives the ensuing characterization of solutions of the equation \(\Delta ^{m}u=0.\)

Theorem 3.3

Let u be a real analytic function defined in a region \(\Omega \subset \mathbb {R}^{n}.\) Let \({\mathbf {a}}\in \Omega .\) Suppose that for all polynomials p of degree k the one dimensional function \(\varphi _{p}\) defined by (3.1) is a polynomial of degree \(k+2m-2\) at the most for \(0\le r<\eta _{p}.\) Then u satisfies the equation

$$\begin{aligned} \Delta ^{m}u=0\,, \end{aligned}$$
(3.11)

in \(\Omega .\) Conversely, if u satisfies (3.11) in \(\Omega \) then at each \({\mathbf {a}}\in \Omega \) the function \(\varphi _{p}=\varphi _{p,{\mathbf {a}}}\) is a polynomial of degree \(k+2m-2\) at the most in the interval \([0,\eta _{{\mathbf {a}}})\) if the ball of center \({\mathbf {a}}\) and radius \(\eta _{{\mathbf {a}}}\) is contained in \(\Omega .\)

Interestingly, Theorem 3.3 holds even if \(m=0.\) We also have the following characterization of solutions of \(\Delta ^{m}u=0.\)

Theorem 3.4

Let u be a real analytic function defined in a region \(\Omega \subset \mathbb {R}^{n}.\) Let \({\mathbf {a}}\in \Omega .\) Suppose that for all polynomials p of degree k

$$\begin{aligned} \mathrm {F.p.}\lim _{r\rightarrow 0^{+}}\frac{\varphi _{p}\left( r\right) }{r^{\alpha }}=0\,,\ \ \ \ \ \text {for all }\alpha >k+2m-2. \end{aligned}$$
(3.12)

Then u satisfies the Eq. (3.11) in \(\Omega .\)

The smooth case

We shall now consider a corresponding characterization of harmonic functions, and of solutions of the equation \(\Delta ^{m}u=0,\) 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 \(B_{\eta }\left( {\mathbf {a}}\right) .\) Suppose that for all polynomials \(p\in {\mathcal {P}}_{k}\) the one dimensional function \(\varphi _{p}\left( r\right) =\int _{\mathbb {S} }u\left( \mathbf {a+}r\omega \right) p\left( \omega \right) \,\mathrm {d}\omega ,\) is a polynomial of degree \(k+2m-2\) at the most for \(0\le r<\eta .\) Then u is a solution of \(\Delta ^{m}u=0\) in \(B_{\eta }\left( {\mathbf {a}}\right) .\)

Proof

In order to better illustrate the ideas, we shall only present the proof in the case \(m=1.\) In other words, we shall show that if \(\varphi _{p}\) is a polynomial of degree k at the most whenever p is, then u is harmonic. Let \(\left\{ {\mathfrak {B}}_{k}\right\} _{k=0}^{\infty }\) be a family of orthonormal basis of the spaces \({\mathcal {H}}_{k},\) \({\mathfrak {B}}_{k}=\left\{ {\mathsf {Y}}_{k,\mu }:\mu \in {\mathfrak {I}}_{k}\right\} ,\) and let us write \(u\left( \mathbf {a+}r\omega \right) \) as a Fourier-Laplace series,

$$\begin{aligned} u\left( \mathbf {a+}r\omega \right) =\sum _{k=0}^{\infty }\sum _{\mu \in {\mathfrak {I}}_{k}}\psi _{k,\mu }\left( r\right) {\mathsf {Y}}_{k,\mu }\left( \omega \right) \,, \end{aligned}$$
(4.1)

a series that converges uniformly on compacts of \(B_{\eta }\left( {\mathbf {a}}\right) .\) As it is well known [9, 18], the functions \(\psi _{k,\mu }\) are smooth in the interval \([0,\eta )\) and actually \(\psi _{k,\mu }\left( r\right) =c_{k,\mu }r^{k}+O\left( r^{k+1}\right) \) for some constant \(c_{k,\mu }\) as \(r\rightarrow 0^{+}.\)

On the other hand, \(\psi _{k,\mu }\) is exactly \(\varphi _{\overline{{\mathsf {Y}} }_{k,\mu }},\) so that it is a polynomial of degree k at the most. Consequently \(\psi _{k,\mu }\left( r\right) =c_{k,\mu }r^{k},\) and thus u equals the sum of harmonic functions

$$\begin{aligned} u\left( {\mathbf {x}}\right) =\sum _{k=0}^{\infty }\sum _{\mu \in {\mathfrak {I}}_{k} }c_{k,\mu }{\mathsf {Y}}_{k,\mu }\left( \mathbf {x-a}\right) \,, \end{aligned}$$
(4.2)

uniformly on compacts of the ball \(B_{\eta }\left( {\mathbf {a}}\right) .\) It follows that u is harmonic in \(B_{\eta }\left( {\mathbf {a}}\right) .\) \(\square \)

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.

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 \(\rho \in {\mathcal {D}}\left( \mathbb {R}^{n}\right) \) be a function that satisfies

$$\begin{aligned} \rho \left( {\mathbf {x}}\right) =1\,,\ \ \ \ \ \ \text {if }\left| {\mathbf {x}}\right| <1\,;\ \ \rho \left( {\mathbf {x}}\right) =2\,,\ \ \ \ \ \ \text {if }\left| {\mathbf {x}}\right| >2\,. \end{aligned}$$
(4.3)

For \(k=1,2,3,\ldots \) let \({\mathsf {Y}}_{k}\in {\mathcal {H}}_{k}\) be a non zero harmonic homogeneous polynomial of degree k. We can then choose an increasing sequence of positive \(\left\{ \lambda _{k}\right\} _{k=1}^{\infty }\) such that the function

$$\begin{aligned} u\left( {\mathbf {x}}\right) =\sum _{k=1}^{\infty }\rho \left( \lambda _{k}{\mathbf {x}}\right) {\mathsf {Y}}_{k}\left( {\mathbf {x}}\right) \,, \end{aligned}$$
(4.4)

that converges for all \({\mathbf {x}},\) since it is a finite sum for \({\mathbf {x}}\ne {\mathbf {0}}\) and a sum of zeros if \({\mathbf {x}}={\mathbf {0}},\) is smooth at the origin. The function u is of course smooth for \({\mathbf {x}} \ne {\mathbf {0}},\) so that, in fact, \(u\in {\mathcal {D}}\left( \mathbb {R} ^{n}\right) .\)

If p is a polynomial of degree k,  then \(\varphi _{p}\left( r\right) \) is a polynomial of degree k at the most if \(0\le r\le 1/\lambda _{k}\) but vanishes for \(r\ge 2/\lambda _{k}.\) Therefore, unless it vanishes, \(\varphi _{p}\) is never a polynomial in the interval \([0,3/\lambda _{k}].\) If \(\eta \) is fixed, then for k large enough, there are polynomials p of degree k such that \(\varphi _{p}\) is not a polynomial of degree k in \([0,\eta ];\) therefore Theorem 3.1 yields that u is not harmonic in \(B_{\eta }\left( {\mathbf {0}}\right) .\)

The distributional case

We shall now consider the characterization of certain distributions that satisfy the polynomial 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 \(f\in {\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) \) be a distribution defined in the ball \(B=B_{\eta }\left( {\mathbf {a}}\right) .\) We say that f has the distributional point value L at \(\mathbf {x=a}\) in the sense of Łojasiewicz [12, 13] if

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}f\left( {\mathbf {a}}+\varepsilon {\mathbf {x}} \right) =L\,, \end{aligned}$$
(5.1)

in \({\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) ,\) that is, if for all test functions \(\phi \in {\mathcal {D}}\left( \mathbb {R}^{n}\right) \)

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\left\langle f\left( {\mathbf {a}}+\varepsilon {\mathbf {x}}\right) ,\phi \left( {\mathbf {x}}\right) \right\rangle =L\int _{\mathbb {R}^{n}}\phi \left( {\mathbf {x}}\right) \,\mathrm {d} {\mathbf {x}}\,. \end{aligned}$$
(5.2)

We use the notation \(L=f\left( {\mathbf {a}}\right) \) \(\left({\L }\right) \) . If all distributions \(\nabla ^{\alpha }f\) have distributional point values at \(\mathbf {x=a}\) for multiindexes \(\left| \alpha \right| \le q,\) then we have the approximation

$$\begin{aligned} f\left( {\mathbf {a}}+\varepsilon {\mathbf {x}}\right) =\sum _{\left| \alpha \right| \le q}\frac{\nabla ^{\alpha }f\left( {\mathbf {a}}\right) }{\alpha !}{\mathbf {x}}^{\alpha }\varepsilon ^{\left| \alpha \right| }+o\left( \varepsilon ^{q}\right) \,,\ \ \ \ \text {as }\varepsilon \rightarrow 0\,, \end{aligned}$$
(5.3)

in \({\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) .\)

Definition 5.1

Let \(f\in {\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) .\) We say that f is distributionally smooth at \(\mathbf {x=a}\) if for all \(\alpha \in \mathbb {N}^{n}\) the Łojasiewicz point values \(\nabla ^{\alpha }f\left( {\mathbf {a}}\right) \) \(\left({\L }\right) \) exist.

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

Lemma 5.2

Let \(f\in {\mathcal {D}}^{\prime }\left( \mathbb {R} ^{n}\right) .\) Then f is distributionally smooth at \(\mathbf {x=a}\) if and only if there exists a smooth function \(\psi \in {\mathcal {E}}\left( \mathbb {R}^{n}\right) \) such that

$$\begin{aligned} f\left( {\mathbf {a}}+\varepsilon {\mathbf {x}}\right) =\psi \left( \varepsilon {\mathbf {x}}\right) +o\left( \varepsilon ^{\infty }\right) \,, \end{aligned}$$
(5.4)

as \(\varepsilon \rightarrow 0\) in the space \({\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) .\)

Proof

If f is distributionally smooth at \(\mathbf {x=a}\) then we may employ Borel’s theorem [6, Thm. 1.5.3] to find a smooth function \(\psi \in {\mathcal {E}}\left( \mathbb {R}^{n}\right) \) such that \(\nabla ^{\alpha }\psi \left( {\mathbf {0}}\right) =\nabla ^{\alpha }f\left( {\mathbf {a}}\right) \) for all \(\alpha \in \mathbb {N}^{n}.\) Then \(\psi \) satisfies (5.4). The converse result is clear. \(\square \)

Next let us consider the polynomial spherical means of distributions. Let \(f\in {\mathcal {D}}^{\prime }\left( B_{\eta }\left( {\mathbf {a}}\right) \right) .\) Let \(p\in {\mathcal {P}}_{k}.\) Then the mean

$$\begin{aligned} \varphi _{p}\left( r\right) =\left\langle f\left( \mathbf {a+}r\omega \right) ,p\left( \omega \right) \right\rangle _{\omega }\,, \end{aligned}$$
(5.5)

is a distribution in the interval \([0,\eta )\) of the space \(\left( r^{k+n-1}{\mathcal {D}}_{\text {even}}[0,\eta )\right) ^{\prime }\) given as

$$\begin{aligned} \left\langle \varphi _{p}\left( r\right) ,\rho \left( r\right) \right\rangle _{r}=\left\langle f\left( \mathbf {a+x}\right) ,\left| {\mathbf {x}} \right| ^{1-k-n}\rho \left( \left| {\mathbf {x}}\right| \right) \right\rangle _{{\mathbf {x}}}\,, \end{aligned}$$
(5.6)

where \(\rho \in r^{k+n-1}{\mathcal {D}}_{\text {even}}[0,\eta ),\) that is, \(\rho \left( r\right) =r^{k+n-1}\tau \left( r\right) \) for \(0\le r<\eta ,\) where \(\tau \) is the restriction to \([0,\eta )\) of an even function of \({\mathcal {D}}(-\eta ,\eta ).\) Formula (5.6) is just the formula for changing variables to polar coordinates in \(\mathbb {R}^{n}.\) Observe also that the notation \({\mathcal {R}}_{k+n-1}\) is used if instead of employing \(r^{k+n-1}{\mathcal {D}}_{\text {even}}\) we use \(r^{k+n-1}{\mathcal {S}} _{\text {even}}\) [8]; in general if \({\mathcal {A}}\left( -\eta ,\eta \right) \) is a space of test functions over an open interval around the origin, then \({\mathcal {A}}[0,\eta )\) denotes the set of restrictions of the elements of \({\mathcal {A}}\left( -\eta ,\eta \right) \) to \([0,\eta )\) [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 \({\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) \) that is distributionally smooth at \(\mathbf {x=a}.\) Let \({\mathsf {Y}}_{k}\in {\mathcal {H}}_{k}.\) Then

$$\begin{aligned} \varphi _{{\mathsf {Y}}}\left( \varepsilon r\right) =\left\langle f\left( \mathbf {a+}\varepsilon r\omega \right) ,{\mathsf {Y}}\left( \omega \right) \right\rangle \sim C\sum _{m=0}^{\infty } \frac{\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) f\right| _{{\mathbf {a}}}}{W_{n,k,m}}r^{k+2m}\varepsilon ^{k+2m}, \end{aligned}$$
(5.7)

as \(\varepsilon \rightarrow 0\) in the space \(\left( r^{k+n-1}{\mathcal {D}} _{\text {even}}[0,\infty )\right) ^{\prime },\) that is, if \(\rho \in r^{k+n-1}{\mathcal {D}}_{\text {even}}[0,\infty )\) then

$$\begin{aligned} \left\langle \varphi _{{\mathsf {Y}}}\left( \varepsilon r\right) ,\rho \left( r\right) \right\rangle \sim C\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) f\right| _{{\mathbf {a}}} }{W_{n,k,m}}\left\langle r^{k+2m},\rho \left( r\right) \right\rangle \varepsilon ^{k+2m}, \end{aligned}$$
(5.8)

as \(\varepsilon \rightarrow 0.\) The formula never holds for all test functions \(\rho \) if we replace \({\mathsf {Y}}\) by a polynomial of \({\mathcal {P}}_{k} \setminus {\mathcal {H}}_{k}.\)

Proof

Let \(\psi \) be a smooth function in \(\mathbb {R}^{n}\) such that \(f\left( {\mathbf {a}}+\varepsilon {\mathbf {x}}\right) =\psi \left( \varepsilon {\mathbf {x}}\right) +o\left( \varepsilon ^{\infty }\right) \) as in (5.4). Then if \(\rho \in r^{k+n-1}{\mathcal {D}}_{\text {even}}[0,\infty ),\) we may apply (2.6) to \(\psi \) to obtain

$$\begin{aligned} \left\langle \left\langle f\left( \mathbf {a+}\varepsilon r\omega \right) ,{\mathsf {Y}}\left( \omega \right) \right\rangle ,\rho \left( r\right) \right\rangle&=\left\langle \int _{\mathbb {S}}\psi \left( \varepsilon r\omega \right) {\mathsf {Y}} \left( \omega \right) \,\mathrm {d}\omega ,\rho \left( r\right) \right\rangle +o\left( \varepsilon ^{\infty }\right) \\&\sim C\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \psi \right| _{{\mathbf {0}}}}{W_{n,k,m}}\left\langle r^{k+2m},\rho \left( r\right) \right\rangle \varepsilon ^{k+2m}, \end{aligned}$$

and (5.8) follows since \(\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \psi \right| _{{\mathbf {0}}}=\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) f\right| _{{\mathbf {a}}}\) for all m. \(\square \)

We shall also need the Fourier-Laplace series of distributions [5]. Let \(f\in {\mathcal {D}}^{\prime }\left( B_{\eta }\left( {\mathbf {0}}\right) \right) .\) Let \(\left\{ {\mathfrak {B}}_{k}\right\} _{k=0}^{\infty }\) be a family of orthonormal basis of the spaces \({\mathcal {H}} _{k},\) \({\mathfrak {B}}_{k}=\left\{ {\mathsf {Y}}_{k,\mu }:\mu \in {\mathfrak {I}} _{k}\right\} .\) Then we can write \(f\left( {\mathbf {x}}\right) \) as a Fourier-Laplace series in the ball \(B_{\eta }\left( {\mathbf {0}}\right) \)

$$\begin{aligned} f\left( r\omega \right) =\sum _{k=0}^{\infty }\sum _{\mu \in {\mathfrak {I}}_{k}}f_{k,\mu }\left( r\right) {\mathsf {Y}}_{k,\mu }\left( \omega \right), \end{aligned}$$
(5.9)

where the \(f_{k,\mu }\) are the distributions \(\left\langle f\left( r\omega \right) ,\overline{{\mathsf {Y}}}\left( \omega \right) \right\rangle \) of the space \(\left( r^{k+n-1}{\mathcal {D}}_{\text {even} }[0,\infty )\right) ^{\prime },\) in the sense of convergence in \({\mathcal {D}} ^{\prime }\left( B_{\eta }\left( {\mathbf {0}}\right) \right) ,\) that is, for all test functions \(\phi \in {\mathcal {D}}\left( B_{\eta }\left( {\mathbf {0}} \right) \right) \) we have

$$\begin{aligned} \left\langle f\left( {\mathbf {x}}\right) ,\phi \left( {\mathbf {x}}\right) \right\rangle =\sum _{k=0}^{\infty }\sum _{\mu \in {\mathfrak {I}}_{k}}\left\langle f_{k,\mu }\left( \left| {\mathbf {x}}\right| \right) {\mathsf {Y}}_{k,\mu }\left( {\mathbf {x}}/\left| {\mathbf {x}}\right| \right) ,\phi \left( {\mathbf {x}}\right) \right\rangle \,. \end{aligned}$$
(5.10)

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

Theorem 5.4

Let u be a distribution of the space \({\mathcal {D}} ^{\prime }\left( B_{\eta }\left( {\mathbf {a}}\right) \right) \) that is distributionally smooth at \(\mathbf {x=a}.\) Suppose that for all polynomials \(p\in {\mathcal {P}}_{k}\) the one dimensional distribution \(\varphi _{p}\left( r\right) =\left\langle u\left( \mathbf {a+}r\omega \right) ,p\left( \omega \right) \right\rangle _{\omega }\) is a regular distribution given by a polynomial of degree \(k+2m-2\) at the most in \([0,\eta ).\) Then u is a solution of \(\Delta ^{m}u=0\) in \(B_{\eta }\left( {\mathbf {a}}\right) .\)

Proof

As before we shall consider the case \(m=1.\) Let us write \(u\left( \mathbf {a+}r\omega \right) \) as a Fourier-Laplace series, \(\sum _{k=0}^{\infty }\sum _{\mu \in {\mathfrak {I}}_{k}}f_{k,\mu }\left( r\right) {\mathsf {Y}}_{k,\mu }\left( \omega \right) .\) Then \(f_{k,\mu }\left( r\right) =\varphi _{\overline{{\mathsf {Y}}}_{k,\mu }}\left( r\right) \) is a regular distribution given by a polynomial of degree k at the most, while because u is distributionally smooth at \({\mathbf {a}},\) the Proposition 5.3 yields that it has a distributional asymptotic behavior of the type \(f_{k,\mu }\left( \varepsilon r\right) =c_{k,\mu }r^{k}\varepsilon ^{k}+O\left( \varepsilon ^{k+1}\right) \) as \(\varepsilon \rightarrow 0\) in the space \(\left( r^{k+n-1}{\mathcal {D}}_{\text {even}}[0,\eta )\right) ^{\prime }.\) Consequently \(f_{k,\mu }\left( r\right) =c_{k,\mu }r^{k}\) and thus

$$\begin{aligned} u\left( {\mathbf {x}}\right) =\sum _{k=0}^{\infty }\sum _{\mu \in {\mathfrak {I}}_{k} }c_{k,\mu }{\mathsf {Y}}_{k,\mu }\left( \mathbf {x-a}\right) \,,\ \ \ \ \text {in }{\mathcal {D}}^{\prime }\left( B_{\eta }\left( {\mathbf {0}}\right) \right) \,. \end{aligned}$$
(5.11)

If we now observe that the space of harmonic functions is a closed subspace of \({\mathcal {D}}^{\prime }\left( B_{\eta }\left( {\mathbf {0}}\right) \right) ,\) we conclude that u is harmonic in \(B_{\eta }\left( {\mathbf {a}}\right) .\) \(\square \)

We must point out that one needs to ask u to be distributionally smooth at \({\mathbf {a}},\) 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 \({\mathsf {Y}}\in {\mathcal {H}}_{m},\) where \(m>1.\) Let q be an integer with \(0<q<m\) and consider the function defined as

$$\begin{aligned} u\left( {\mathbf {x}}\right) =\left| {\mathbf {x}}\right| ^{q} {\mathsf {Y}}\left( \frac{{\mathbf {x}}}{\left| {\mathbf {x}}\right| }\right) \,, \end{aligned}$$
(5.12)

for \(\mathbf {x\ne 0}\) and \(u\left( {\mathbf {0}}\right) =0.\) Then u is continuous, actually of class \(C^{q-1},\) and \(\int _{\mathbb {S}}u\left( r\omega \right) p\left( \omega \right) \,\mathrm {d} \omega =0\) if \(p\in {\mathcal {H}}_{k}\) and \(k<m,\) while, if it does not vanish, it is a polynomial of degree q when \(k\ge m.\) Naturally u is not harmonic.

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 \(\Omega ,\) then for all subregions \(\Lambda \) with a smooth boundary we clearly have the zero flux property

$$\begin{aligned} \int _{\partial \Lambda }\frac{\partial u}{\partial n}\left( \xi \right) \,\mathrm {d}\xi =0\,, \end{aligned}$$
(6.1)

whereFootnote 2 \(\partial u/\partial n=u_{,i}n_{i}\) denotes the exterior normal derivative of u on the boundary \(\partial \Lambda .\) A converse result was already given in 1906 by Bôcher and Koebe, who independently proved that if

$$\begin{aligned} \int _{\partial B}\frac{\partial u}{\partial n}\left( \xi \right) \,\mathrm {d}\xi =0\,, \end{aligned}$$
(6.2)

for all balls contained in \(\Omega \) 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

$$\begin{aligned} \chi _{p}\left( r\right) =\int _{\mathbb {S}}\frac{\partial u}{\partial n}\left( {\mathbf {a}}+r\omega \right) \,p\left( \omega \right) \,\mathrm {d}\omega \,, \end{aligned}$$
(6.3)

around a fixed point \({\mathbf {a}}\) for all polynomials p.

Our main tool is the following asymptotic formula.

Proposition 6.1

Let \(\phi \) be a smooth function in \(\mathbb {R}^{n}.\) Then if \({\mathsf {Y}}\in {\mathcal {H}}_{k},\)

$$\begin{aligned} \int _{\mathbb {S}}\phi _{,i}\left( \varepsilon \omega \right) \omega _{i}\,{\mathsf {Y}}\left( \omega \right) \,\mathrm {d} \omega \sim C\sum _{m=1}^{\infty }\frac{\left( 2m+k\right) \left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \phi \right| _{{\mathbf {0}}}}{W_{n,k,m}}\varepsilon ^{k+2m-1}, \end{aligned}$$
(6.4)

as \(\varepsilon \rightarrow 0.\)

Proof

Let \(\psi \left( {\mathbf {x}}\right) =x_{i}\phi _{,i}\left( {\mathbf {x}}\right) =\left( D\phi \right) \left( {\mathbf {x}}\right) ,\) where \(D=x_{i}\nabla _{i}\) is Euler’s differential operator. Then the asymptotic formula (2.6) yields

$$\begin{aligned} \int _{\mathbb {S}}\phi _{,i}\left( \varepsilon \omega \right) \omega _{i}\,{\mathsf {Y}}\left( \omega \right) \,\mathrm {d} \omega&=\frac{1}{\varepsilon }\int _{\mathbb {S}}\psi \left( \varepsilon \omega \right) \,{\mathsf {Y}}\left( \omega \right) \,\mathrm {d}\omega \\&\sim C\sum _{m=0}^{\infty }\frac{\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \psi \right| _{{\mathbf {0}}}}{W_{n,k,m}} \varepsilon ^{k+2m-1}. \end{aligned}$$

If we now observe that for all multi-indices \(\alpha \)

$$\begin{aligned} \nabla ^{\alpha }D=\left| \alpha \right| \nabla ^{\alpha }+D\nabla ^{\alpha }, \end{aligned}$$
(6.5)

we obtain \(\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \psi \right| _{{\mathbf {0}}}=\left( 2m+k\right) \left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \phi \right| _{{\mathbf {0}}},\) and (6.4) follows. \(\square \)

The case \({\mathsf {Y}}=1\) is already interesting, since the formula

$$\begin{aligned} \int _{\mathbb {S}}\phi _{,i}\left( \varepsilon \omega \right) \omega _{i}\,\mathrm {d}\omega \sim 2C\sum _{m=1}^{\infty }\frac{m\left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) \phi \right| _{{\mathbf {0}}}}{W_{n,k,m}}\varepsilon ^{2m-1}, \end{aligned}$$
(6.6)

allows us to obtain Saks’ 1932 characterization of harmonic functions [11], namely, if u is smoothFootnote 3 and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon }\int _{\mathbb {S}} u_{,i}\left( {\mathbf {a}}+\varepsilon \omega \right) \omega _{i}\,\mathrm {d}\omega =0\,, \end{aligned}$$
(6.7)

for all \({\mathbf {a}}\in \Omega ,\) then u is harmonic in \(\Omega .\)

Theorem 6.2

Let u be a real analytic function defined in a region \(\Omega \subset \mathbb {R}^{n}.\) Let \({\mathbf {a}}\in \Omega .\) Suppose that for all polynomials p of degree k the function of one variable \(\chi _{p}\) given by (6.3) satisfies

$$\begin{aligned} \mathrm {F.p.}\lim _{r\rightarrow 0^{+}}\frac{\chi _{p}\left( r\right) }{r^{\alpha }}=0\,,\ \ \ \ \ \text {for all }\alpha >k-1. \end{aligned}$$
(6.8)

Then u is harmonic in \(\Omega .\) In particular, if for all \(p\in {\mathcal {P}}_{k}\) there exists \(r_{p}>0\) such that \(\chi _{p}\left( r\right) \) is a polynomial of degree \(k-1\) at the most for \(0\le r\le r_{p},\) then u is harmonic in \(\Omega .\)

Proof

We just need to make obvious changes in the proof of the Theorem 3.2. \(\square \)

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 \({\mathcal {D}}^{\prime }\left( \mathbb {R}^{n}\right) \) that is distributionally smooth at \(\mathbf {x=0}.\) Let \({\mathsf {Y}}\in {\mathcal {H}}_{k}.\) Then

$$\begin{aligned} \left\langle \nabla _{i}f\left( \varepsilon r\omega \right) ,\omega _{i}{\mathsf {Y}}\left( \omega \right) \right\rangle \sim C\sum _{m=1}^{\infty }\frac{\left( 2m+k\right) \left. \Delta ^{m} {\mathsf {Y}}\left( \nabla \right) f\right| _{{\mathbf {0}}} }{W_{n,k,m}}\varepsilon ^{k+2m-1}r^{k+2m-1}, \end{aligned}$$
(6.9)

as \(\varepsilon \rightarrow 0\) in the space \(\left( r^{k+n}{\mathcal {D}} _{\text {even}}[0,\infty )\right) ^{\prime },\) that is, if \(\rho \in r^{k+n}{\mathcal {D}}_{\text {even}}[0,\infty )\) then

$$\begin{aligned} \left\langle \left\langle \nabla _{i}f\left( \varepsilon r\omega \right) ,\omega _{i}{\mathsf {Y}}\left( \omega \right) \right\rangle ,\rho \left( r\right) \right\rangle \sim C\sum _{m=1}^{\infty }\frac{\left( 2m+k\right) \left. \Delta ^{m}{\mathsf {Y}}\left( \nabla \right) f\right| _{{\mathbf {0}}}}{W_{n,k,m}}\left\langle r^{k+2m-1},\rho \left( r\right) \right\rangle \varepsilon ^{k+2m-1}, \end{aligned}$$
(6.10)

as \(\varepsilon \rightarrow 0.\)

We immediately obtain the ensuing result.

Theorem 6.4

Let u be a distribution of the space \({\mathcal {D}} ^{\prime }\left( B_{\eta }\left( {\mathbf {a}}\right) \right) \) that is distributionally smooth at \(\mathbf {x=a}.\) Suppose that for all polynomials \(p\in {\mathcal {P}}_{k}\) the one dimensional distribution \(\chi _{p}\left( r\right) =\left\langle \nabla _{i}u\left( \mathbf {a+}r\omega \right) ,\omega _{i}p\left( \omega \right) \right\rangle _{\omega }\) is a regular distribution given by a polynomial of degree \(k-1\) at the most in \([0,\eta ).\) Then u is harmonic in \(B_{\eta }\left( {\mathbf {a}}\right) .\)

Notes

  1. 1.

    For example the expression \(\omega _{1}+\omega _{2}^{3}\) does not look homogeneous, but corresponds to an element of \({\mathcal {P}}_{3},\) namely \(x_{1}r^{2} +x_{2}^{3}.\)

  2. 2.

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

  3. 3.

    Using the arguments of [4] we can just assume that \(u_{,i}\) are locally integrable.

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Estrada, R. Characterization of harmonic functions by the behavior of means at a single point. SN Partial Differ. Equ. Appl. 1, 2 (2020). https://doi.org/10.1007/s42985-019-0003-z

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Keywords

  • Harmonic functions
  • Harmonic polynomials
  • Mean value theorems

Mathematics Subject Classification

  • Primary 31B05
  • 33C55
  • 35B05; Secondary 46F10