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Three spheres inequality

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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

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Let be an harmonic function on . Then for all one has

where for is the sphere of radius centred at the origin and where

Here we use the following normalisation for the norm:

References

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  • 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