$\mathcal {R}_{s}$ -Sectorial Operators and Generalized Triebel-Lizorkin Spaces

Abstract

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.