On maximal regularity and semivariation of $\alpha$-times resolvent families

TL;DR

This paper investigates the conditions under which fractional Cauchy problems involving $eta$-times resolvent families exhibit maximal regularity, linking it to the bounded semivariation of the associated resolvent family.

Contribution

It establishes a necessary and sufficient condition for maximal regularity of fractional Cauchy problems based on the semivariation of $eta$-times resolvent families.

Findings

Maximal regularity holds iff the resolvent family has bounded semivariation.

Provides a characterization of fractional Cauchy problem regularity.

Connects fractional calculus with operator semigroup theory.

Abstract

Let $1< \alpha <2$ and $A$ be the generator of an $\alpha$-times resolvent family $\{S_\alpha(t)\}_{t \ge 0}$ on a Banach space $X$. It is shown that the fractional Cauchy problem ${\bf D}_t^\alpha u(t) = Au(t)+f(t)$, $t \in [0,r]$; $u(0), u'(0) \in D(A)$ has maximal regularity on $C([0,r];X)$ if and only if $S_\alpha(\cdot)$ is of bounded semivariation on $[0,r]$.