Three spheres inequality

In mathematics, the three spheres inequality bounds the ${\displaystyle L^{2}}$ norm of an 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.

Statement of the three spheres inequality

Let ${\displaystyle u}$ be an harmonic function on ${\displaystyle \mathbb {R} ^{n}}$. Then for all ${\displaystyle 0<r_{1}<r<r_{2}}$ one has

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

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

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

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