Sobolev Type Spaces Based on Lorentz-Karamata Spaces

Abstract

In this paper, firstly Lorentz-Karamata-Sobolev spaces $W_{L\left(p,q;b\right) }^{k}\left( \mathbb{R}^{n}\right) $ of integer order are introduced and some of their important properties are emphasized. Also, Banach spaces $A_{L\left( p,q;b\right) }^{k}\left(\mathbb{R}^{n}\right) =L^{1}\left(\mathbb{R}^{n}\right) $ $\cap W_{L\left( p,q;b\right) }^{k}\left(\mathbb{R}^{n}\right) $ (Lorentz-Karamata-Sobolev algebras) are studied. Using a result of H.C.Wang, it is showed that Banach convolution algebras $A_{L\left( p,q;b\right) }^{k}\left(\mathbb{R}^{n}\right) $ don't have weak factorization and the multiplier algebra of $A_{L\left( p,q;b\right) }^{k}\left(\mathbb{R}^{n}\right) $ coincides with the measure algebra $M\left(\mathbb{R}^{n}\right) $ for $1<p<\infty $ and $1\leq q<\infty $.