Besov spaces with variable smoothness and integrability

Function Spaces of Variable Smoothness and Integrability

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 ...

Sharp Embeddings of Besov Spaces with Logarithmic Smoothness

We prove sharp embeddings of Besov spaces B p,r (R ) with the classical smoothness σ and a logarithmic smoothness α into Lorentz-Zygmund spaces. Our results extend those with α = 0, which have been proved by D. E. Edmunds and H. Triebel. On page 88 of their paper (Math. Nachr. 207 (1999), 79–92) they have written: “Nevertheless a direct proof, avoiding the machinery of function spaces, would be...

Approximation and entropy numbers in Besov spaces of generalized smoothness

Let X and Y be Banach (or quasi-Banach) spaces and let T ∈ L(X,Y ) be a compact operator. In order to measure “the degree of compactness of T” one can use the asymptotic decay of the sequence of approximation numbers of T or the decay of the sequence of entropy numbers of T. In general, the behaviour of these sequences is quite different. However, as we shall show, under certain assumptions the...

Local Growth Envelopes of Besov Spaces of Generalized Smoothness

The concept of local growth envelope (ELGA, u) of the quasi-normed function space A is applied to the Besov spaces of generalized smoothness B p,q (Rn).

Pointwise Multipliers of Besov Spaces of Smoothness Zero and Spaces of Continuous Functions