Paley-Littlewood decomposition for sectorial operators and interpolation spaces

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, Schrödinger 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.

2nd version to appear in Mathematische Nachrichten, Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2016

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• Euclidean structures and operator theory in Banach spaces
• Oversmoothing Tikhonov regularization in Banach spaces
• Almost everywhere convergence of Bochner-Riesz means on Heisenberg-type groups
• Maximal Hörmander Functional Calculus on Lp Spaces and UMD Lattices
• Spectral multiplier theorems via $H^{\infty }$ calculus and $R$-bounds
• Variational source conditions in Lp-spaces
• Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager's conjecture in fluid dynamics
• A Besov algebra calculus for generators of operator semigroups and related norm-estimates
• Besov spaces associated with non-negative operators on Banach spaces