A note on polyharmonic functions

In this note we prove the following theorem: Suppose 0−1. 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.  2003 Elsevier Science (USA). All rights reserve