In mathematics, the three spheres inequality bounds the
norm of a harmonic function on a given sphere in terms of the
norm of this function on two spheres, one with bigger radius and one with smaller radius.
Statement of the three spheres inequality
Let
be an harmonic function on
. Then for all
one has
![{\displaystyle \|u\|_{L^{2}(S_{r})}\leq \|u\|_{L^{2}(S_{r_{1}})}^{\alpha }\|u\|_{L^{2}(S_{r_{2}})}^{1-\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9292c7924d2b1ec48a909ceb51562d1ff7338ed9)
where
for
is the sphere of radius
centred at the origin and where
![{\displaystyle \alpha :={\frac {\log(r_{2}/r)}{\log(r_{2}/r_{1})}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee4831e68ea513fc0985d5f2336a5fc09f48358)
Here we use the following normalisation for the
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}}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e64f119ea80241a882d2029b51350a4a074338d)
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