A note on polyharmonic functions

Abstract

In this note we prove the following theorem: Suppose 0 < p < ∞ and α > - 1. Then there is a constant C = C(p, m, n, α) such that ∫B u(x) p(1 - x )α dV(x) ≤ C( u(0) p + ∫B ∇u(x) p(1 - x )p+α dV(x)), for all polyharmonic functions u of order m, on the unit ball B ⊂ Rn.