Generalizations of inequalities of littlewood and paley

Abstract

For a function f, holomorphic in the open unit ball Bn in Cn, with f(0)=0, we prove (I) If 0<s≤2 and s≤p<∞ Then ‖f‖pp≤C∫01∫∂Bn|f(ρζ)|p−s|Rf(ρζ)|s(log1/ρ)s−1ρ−1dσ(ζ)dρ (ii) If 2≤B≤p<∞ Then ∫01∫∂Bn|f(ρζ)|p−s|Rf(ρζ)|s(log1/ρ)s−1ρ−1dσ(ζ)dρ≤C‖f‖pp where Rf is the radial dervative of f, generalizing the known cases p=s([1]) and p=s, n=1 ([2]).