A. In this article we introduce Triebel–Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Applying it, we give molecular ...

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, the theory electrorheological fluids as well image processing regions where p ( x ) reaches value 1.

In this paper, direct estimate of Sobolev exponent of reenable distributions and its application to the asymptotic estimate of Sobolev exponent of M band Daubechies' scaling functions are considered.