Weighted integrals for polyharmonic type functions

Abstract

In this note we investigate the relationship between the integrals: ∫ B |D β u(x)| p |D γ u(x) | q ψ p|β|+q|γ| (x)ω(x)dV(x); ∫ B |u(x)| p+q ω(x)dV(x) where p, q > 0, β, γ are arbitrary multi-indices and ω is a standard admissible weight with distortion function ψ, for polyharmonic type functions u on the unit ball B.