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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.
Preliminary Functional Analytic Matters
Abstract Fredholm Theory
Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli
Hardy Spaces on Ahlfors Regular Sets
Banach Function Spaces, Extrapolation, and Orlicz Spaces
Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals on Ahlfors Regular Sets
Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets
Boundary Traces from Weighted Sobolev Spaces into Besov Spaces
Besov and Triebel-Lizorkin Spaces in Open Sets
Strong and Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces
Sobolev Spaces on the Geometric Measure Theoretic Boundary of Sets of Locally Finite Perimeter