Three spheres inequality explained

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\| u \| _{L ^{2 (S} _{r)}} \leq \| u \| ^{\alpha} _{L ^{2 (S} _{r _{1}} )} \| u \| ^{1- \alpha} _{L ^{2 (S} _{r _{2}} )} where S _{\rho} : = \{ x \in R ^{n} \colon \vert x \vert = \rho\} for \rho> 0 is the sphere of radius \rho centred at the origin and where\alpha: = (log (r _{2/r)})/(log (r _{2/r} _{1)}) . Here we use the following normalisation for the L ^{2} norm:\| u \| ^{2} _{L ^{2 (S} _{\rho)}} : = \rho ^{1-n} \int _{S ^{n-1}} \vert u(\rho \hat x) \vert ^{2} d\sigma(\hat x).