Three spheres inequality

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

Statement of the three spheres inequality

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

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

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

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

Here we use the following normalisation for the $L^2$ norm:

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

References

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