Geometric Harmonic Analysis II: Function Spaces Measuring Size and Smoothness on Rough Sets

Description

This monograph is part of a larger program, materializing in five volumes, whose principal aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. Volume II is concerned with function spaces measuring size and/or smoothness, such as Hardy spaces, Besov spaces, Triebel-Lizorkin spaces, Sobolev spaces, Morrey spaces, Morrey-Campanato spaces, spaces of functions of Bounded Mean Oscillations, etc., in general geometric settings. Work here also highlights the close interplay between differentiability properties of functions and singular integral operators. The text is intended for researchers, graduate students, and industry professionals interested in harmonic analysis, functional analysis, geometric measure theory, and function space theory. 1 Prefacing the Full Series Description of Volume II Contents 978-3-031-13718-1_1 Preliminary Functional Analytic Matters Algebraic and Operator Theoretic Rudiments Quasi-Normed Spaces Real Interpolation of Quasilinear Operators Complex Interpolation on Quasi-Banach Spaces Köthe Function Spaces and Other Categories of Topological Vector Spaces 978-3-031-13718-1_2 Abstract Fredholm Theory Fredholm Theory in Banach Spaces Fredholm Theory in Topological Vector Spaces 978-3-031-13718-1_3 Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli Functions of Vanishing Mean Oscillations on Measure Metric Spaces A New Brand of Hölder Spaces 978-3-031-13718-1_4 Hardy Spaces on Ahlfors Regular Sets The Fefferman-Stein Grand Maximal Function Lebesgue-Based and Lorentz-Based Hardy Spaces on Ahlfors Regular Sets Real Interpolation of Hardy Spaces Atomic Decompositions for Hardy Spaces Molecules in Hardy Spaces Duality in Hardy Spaces Richness of the Duals of Lorentz-Based Hardy Spaces Weak- Convergence and More on the Compatibility of Pairings More on Hp Versus Lp: The Filtering Operator 978-3-031-13718-1_5 Banach Function Spaces, Extrapolation, and Orlicz Spaces Generalized Banach Function Spaces Extrapolation Theory Young Functions and Orlicz Spaces 978-3-031-13718-1_6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals on Ahlfors Regular Sets Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets 978-3-031-13718-1_7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets Definitions with Sharp Ranges of Indices and Basic Results Atomic and Molecular Theory Calderón's Reproducing Formula and Frame Theory Interpolation of Besov and Triebel-Lizorkin Spaces via the Real Method Complex Interpolation of Besov and Triebel-Lizorkin Spaces Duality Results for Besov and Triebel-Lizorkin Spaces Loose and Tight Embeddings Envelopes of Besov and Triebel-Lizorkin Spaces Intrinsic Characterizations of Besov Spaces 978-3-031-13718-1_8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces Traces from Weighted Sobolev Spaces Defined in Rn Technical Lemmas Traces from Weighted Sobolev Spaces Defined in Open Subsets of Rn Extension Operators from Boundary Besov Spaces into Weighted Sobolev Spaces Weighted Sobolev Spaces of Negative Order, Duality, and the Conormal Derivative Operator Traces from Weighted Maximal Sobolev Spaces in Open Subsets of Rn 978-3-031-13718-1_9 Besov and Triebel-Lizorkin Spaces in Open Sets Besov and Triebel-Lizorkin Spaces in Rn Besov and Triebel-Lizorkin Spaces in Open Subsets of Rn Envelopes of Besov and Triebel-Lizorkin Spaces in Open Sets Traces of Functions from Besov and Triebel-Lizorkin Spaces The Conormal Derivative Operator on Besov and Triebel-Lizorkin Spaces Extension from Boundary Besov Spaces into Smoothness Spaces in Open Sets 978-3-031-13718-1_10 Strong and Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces Strong Normal Boundary Traces of Vector Fields in Hardy Spaces Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces The Manifold Setting 978-3-031-13718-1_11 Sobolev Spaces on the Geometric Measure Theoretic Boundary of Sets of Locally Finite Perimeter Weak Tangential Derivatives and Boundary Sobolev Spaces Distributional Tangential Derivatives Distributional Derivatives Versus Weak Tangential Derivatives The Tangential Gradient Sobolev Spaces on Boundaries of Ahlfors Regular Domains … Boundary Sobolev Spaces on Manifolds A General Recipe for Sobolev-Like Spaces Boundary Sobolev Spaces of Negative Smoothness Principal-Value Distributions on Countably Rectifiable Sets Hardy-Based Boundary Sobolev Spaces Embedding Results Involving Boundary Sobolev Spaces Real Interpolation Involving Boundary Sobolev Spaces Morrey-Based, and Block-Based, Homogeneous Sobolev Spaces 1 (1) Terms and Notation used in Volume II References