Three spheres inequality

In mathematics, the three spheres inequality bounds the [math]\displaystyle{ L^2 }[/math] norm of a harmonic function on a given sphere in terms of the [math]\displaystyle{ L^2 }[/math] norm of this function on two spheres, one with bigger radius and one with smaller radius.

Statement of the three spheres inequality

Let [math]\displaystyle{ u }[/math] be an harmonic function on [math]\displaystyle{ \mathbb R^n }[/math]. Then for all [math]\displaystyle{ 0 \lt r_1 \lt r \lt r_2 }[/math] one has

[math]\displaystyle{ \| u \|_{L^2(S_r)} \leq \| u \|^\alpha_{L^2(S_{r_1})} \| u \|^{1-\alpha}_{L^2(S_{r_2})} }[/math]

where [math]\displaystyle{ S_\rho := \{ x \in \mathbb R^n \colon \vert x \vert = \rho\} }[/math] for [math]\displaystyle{ \rho\gt 0 }[/math] is the sphere of radius [math]\displaystyle{ \rho }[/math] centred at the origin and where

[math]\displaystyle{ \alpha:=\frac{\log(r_2/r)}{\log(r_2/r_1)}. }[/math]

Here we use the following normalisation for the [math]\displaystyle{ L^2 }[/math] norm:

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

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