Three Spheres Inequality

In mathematics, the three spheres inequality bounds the ${\displaystyle L^2}$ norm of a harmonic function on a given sphere in terms of the ${\displaystyle L^2}$ norm of this function on two spheres, one with bigger radius and one with smaller radius.

Let ${\displaystyle u}$ be an harmonic function on ${\displaystyle {\mathbb{ℝ}}^n}$. Then for all ${\displaystyle 0<r_1<r<r_2}$ one has

${\displaystyle \Vert u{\Vert }_{L^2(S_r)}\le \Vert u{\Vert }_{L^2(S_{r_1})}^{\alpha }\Vert u{\Vert }_{L^2(S_{r_2})}^{1-\alpha }}$

where ${\displaystyle S_{\rho }:=\{x\in {\mathbb{ℝ}}^n:|x|=\rho \}}$ for ${\displaystyle \rho >0}$ is the sphere of radius ${\displaystyle \rho }$ centred at the origin and where

${\displaystyle \alpha :=\frac{\log (r_2/r)}{\log (r_2/r_1)}.}$

Here we use the following normalisation for the ${\displaystyle L^2}$ norm:

${\displaystyle \Vert u{\Vert }_{L^2(S_{\rho })}^2:={\rho }^{1-n}\underset{{\mathbb{𝕊}}^{n-1}}{\int }|u(\rho \hat{x}){|}^{2}d\sigma (\hat{x}).}$

• Korevaar, J.; Meyers, J. L. H. (1994), "Logarithmic convexity for supremum norms of harmonic functions", Bull. London Math. Soc. 26 (4): 353–362, doi:10.1112/blms/26.4.353 

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