Three spheres inequality

In mathematics, the three spheres inequality bounds the L2 norm of a harmonic function on a given sphere in terms of the L2 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 ℝn. Then for all 0<r1<r<r2 one has

‖u‖L2(Sr)≤‖u‖L2(Sr1)α‖u‖L2(Sr2)1−α

where Sρ:={x∈ℝn:|x|=ρ} for ρ>0 is the sphere of radius ρ centred at the origin and where

α:=log⁡(r2/r)log⁡(r2/r1).

Here we use the following normalisation for the L2 norm:

‖u‖L2(Sρ)2:=ρ1−n∫𝕊n−1|u(ρx^)|2dσ(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

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