International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 205-207
doi:10.1155/S0161171293000250
Generalizations of inequalities of littlewood and paley
Mathematics Department, Qufu Normal University, Shandong, Qufu 273165, China
Received 30 July 1991; Revised 13 December 1991
Copyright © 1993 Lou Zengjian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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]).