Extensions of Hardy-Littlewood inequalities

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]).