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 $\mathbb {R} ^{n}$ . Then for all $0<r_{1}<r<r_{2}$ one has

\|u\|_{L^{2}(S_{r})}\leq \|u\|_{L^{2}(S_{r_{1}})}^{\alpha }\|u\|_{L^{2}(S_{r_{2}})}^{1-\alpha }

where $S_{\rho }:=\{x\in \mathbb {R} ^{n}\colon \vert x\vert =\rho \}$ 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:

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

References

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, MR 1302068

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