Some results on variable Gaussian Besov-Lipschitz and variable Gaussian Triebel-Lizorkin spaces

Abstract

In a previous paper two of the authors introduced and study Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha }({\gamma }_d)$ and Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha }({\gamma }_d)$. Now, in this paper we introduce the variable Gaussian Besov-Lipschitz spaces $B_{p(\cdot ),q(\cdot )}^{\alpha }({\gamma }_d)$ and the variable Gaussian Triebel-Lizorkin spaces $F_{p(\cdot ),q(\cdot )}^{\alpha }({\gamma }_d),$ that is to say, Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with variable exponents, under certain additional regularity conditions on the exponents $p(\cdot )$ and $q(\cdot )$ introduced by Dalmasso and Scotto. Trivially, they include the Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha }({\gamma }_d)$ and Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha }({\gamma }_d)$. We consider some inclusion relations of those spaces and finally we also prove some interpolation results for them.

22 pages