In mathematics, the three spheres inequality bounds the L 2 {\displaystyle L^{2}} norm of an harmonic function on a given sphere in terms of the L 2 {\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 u {\displaystyle u} be an harmonic function on R n {\displaystyle \mathbb {R} ^{n}} . Then for all 0 < r 1 < r < r 2 {\displaystyle 0<r_{1}<r<r_{2}} one has
‖ u ‖ L 2 ( S r ) ≤ ‖ u ‖ L 2 ( S r 1 ) α ‖ u ‖ L 2 ( S r 2 ) 1 − α {\displaystyle \|u\|_{L^{2}(S_{r})}\leq \|u\|_{L^{2}(S_{r_{1}})}^{\alpha }\|u\|_{L^{2}(S_{r_{2}})}^{1-\alpha }}
where S ρ := { x ∈ R n : | x | = ρ } {\displaystyle S_{\rho }:=\{x\in \mathbb {R} ^{n}\colon \vert x\vert =\rho \}} for ρ > 0 {\displaystyle \rho >0} is the sphere of radius ρ {\displaystyle \rho } centred at the origin and where
α := log ( r 2 / r ) log ( r 2 / r 1 ) . {\displaystyle \alpha :={\frac {\log(r_{2}/r)}{\log(r_{2}/r_{1})}}.}
Here we use the following normalisation for the L 2 {\displaystyle L^{2}} norm:
‖ u ‖ L 2 ( S ρ ) 2 := ρ 1 − n ∫ S n − 1 | u ( ρ x ^ ) | 2 d σ ( x ^ ) . {\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