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

  • Ricardo EstradaEmail author
Original Paper
Part of the following topical collections:
  1. Theory of PDEs


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.


Harmonic functions Harmonic polynomials Mean value theorems 

Mathematics Subject Classification

Primary 31B05 33C55 35B05; Secondary 46F10 


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Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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