Lorentz–Karamata spaces, Bessel and Riesz potentials and embeddings
Volume 405 / 2002
Abstract
We consider Lorentz–Karamata spaces and establish embedding theorems (some local and some global) for Bessel-potential spaces modelled upon appropriate Lorentz–Karamata spaces into Lorentz–Karamata spaces and Orlicz spaces. In particular, we obtain refinements of the Sobolev embedding theorems: Strichartz–Trudinger's limiting case and Hansson–Brézis–Wainger's limiting case. These results extend and improve those of Edmunds, Gurka and Opic. Estimates for an appropriate norm of the convolution of a function in a Lorentz space with one in a Lorentz-Karamata space and suitably restricted at infinity are presented, and consequently also those for the Riesz potential of an appropriate function. The results extend those of Brézis–Wainger on the convolution of functions in Lorentz spaces which lead to exponential integrability and those of Edmunds, Gurka and Opic on double exponential integrability of convolution operators. Furthermore, in some cases the results of Edmunds, Gurka and Opic are improved in the sense that we obtain a triple exponential Orlicz space as a target instead of a double one. Moreover, we also obtain results related to that of Mizuta and Shimomura.
