Paley-Littlewood decomposition for sectorial operators and interpolation spaces

TL;DR

This paper establishes Paley-Littlewood decompositions for fractional powers of sectorial operators, linking them to Triebel-Lizorkin and Besov spaces, with applications to various differential operators and spectral theories.

Contribution

It introduces new Paley-Littlewood decompositions for sectorial operators using $H^$-calculus and spectral multipliers, extending classical harmonic analysis tools to Banach space operators.

Findings

Decomposition results for fractional powers of sectorial operators

Applications to Laplace operators on manifolds and graphs

Extensions to bisectorial operators and group generators

Abstract

We prove Paley-Littlewood decompositions for the scales of fractional powers of $0$-sectorial operators $A$ on a Banach space which correspond to Triebel-Lizorkin spaces and the scale of Besov spaces if $A$ is the classical Laplace operator on $L^p(\mathbb{R}^n).$We use the $H^\infty$-calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace-type operators on manifolds and graphs, Schr\"odinger operators and Hermite expansion.We also give variants of these results for bisectorial operators and for generators of groups with a bounded $H^\infty$-calculus on strips.