International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 1, Pages 193-195
doi:10.1155/S016117129400027X
Extensions of Hardy-Littlewood inequalities
Mathematics Department, Qufu Normal University, Shandong, Qufu 273165, China
Received 30 December 1992; Revised 13 May 1993
Copyright © 1994 Zengjian Lou. 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∈Hp(Bn), with f(0)=0, we prove
(1) If 0<p≤s,then∫01r−1(log1r)sβ−1Mps(r,Rβf)dr≤‖f‖ps−p‖f‖p,s,βp(2) If s≤p<∞, then‖f‖p,s,βp≤‖f‖pp−s∫01r−1(log1r)sβ−1Mps(r,Rβf)drwhere Rβf is the fractional derivative of f. These results generalize the known cases s=2, β=1 ([1]).