Sobolev Type Spaces Based on Lorentz--Karamata Spaces

İlker Eryilmaz

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