An extension of Hedberg's convolution inequality and applications
In his celebrated paper [Proc. Am. Math. Soc. 36, 505-510 (1993; Zbl 0283.26003)], \textit{L. I. Hedberg} presented a new proof of the Sobolev inequality for potentials, which asserts, roughly speaking, that the Riesz potential of order \(\alpha\in(0,n)\) is bounded from \(L^p(\mathbb R^n)\) into \(L^q(\mathbb R^n)\) when \(p\in(1,n/\alpha)\) and \(q=np/(n-\alpha p)\). Hedberg's method is based on a pointwise inequality linking together the Riesz potential and the Hardy-Littlewood maximal operator. The paper under review presents an Orlicz-space version of Hedberg's pointwise inequality and its very interesting applications. In particular, sharp embeddings of Orlicz-Sobolev spaces into Orlicz spaces are obtained. Examples including the limiting case \(p=n/\alpha\) are discussed.
scientific article; zbMATH DE number 1236168
scientific article