Besov spaces associated with non-negative operators on Banach spaces

TL;DR

This paper develops a theory of Besov spaces linked to non-negative operators on Banach spaces, using dyadic resolvent decompositions to connect smoothness with fractional power boundedness.

Contribution

It introduces a new framework for Besov spaces associated with operators, utilizing dyadic resolvent decompositions and establishing links to fractional power boundedness.

Findings

Constructed Besov spaces via dyadic resolvent decomposition.

Established connections between smoothness and fractional power boundedness.

Provided explicit quasi-norm estimates for these Besov spaces.

Abstract

Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces $B^{ s, A }_{ q, X }$ for non-negative operators $A$ on Banach spaces $X$ with a full range of indices $s \in \mathbb{R}$ and $0 < q \leq \infty$. The approach we use is the dyadic decomposition of resolvents for non-negative operators, an analogue of the Littlewood-Paley decomposition in the construction of the classical Besov spaces. In particular, by using the reproducing formulas for fractional powers of operators and explicit quasi-norms estimates for Besov spaces we discuss the connections between the smoothness of Besov spaces associated with operators and the boundedness of fractional powers of the underlying operators.