R-boundedness of smooth operator-valued functions

TL;DR

This paper investigates conditions under which families of operator-valued functions are R-bounded, focusing on Banach space properties, integral operators, and applications to semigroups and stochastic Cauchy problems.

Contribution

It provides new sufficient conditions for R-boundedness of operator families based on cotype, type, and Besov space regularity, with applications to evolution equations.

Findings

Integral operators are R-bounded under certain conditions.

R-boundedness of operator families in Besov spaces is established.

Applications to semigroups and stochastic Cauchy problems are demonstrated.

Abstract

In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems.