On certain convolution inequalities

Abstract

It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers.We also improve a recent extension of Trudinger's inequality due to Strichartz. JjlJl!The following theorem is due to Hardy and Littlewood [3] for d= 1, and to Sobolev [8] in the general case.A simple proof is given in [9, V.l.2].Theorem 1.Let 0<a<d, l<^<^<co, and \lq=\jp-a/d.ThenWÁDhúAWfWr,.Iffis supported by a ball B, and l/9=l-a/d, then Iaif) e L"iB) if$B l/l log+ l/l dx< oo.We first prove a simple lemma.