Search

This monograph is a testament to the potency of the method of singular integrals of layer potential type in solving boundary value problems for weakly elliptic systems in the setting of Muckenhoupt-weighted Morrey spaces and their pre-duals. A functional analytic framework for Muckenhoupt-weighted Morrey spaces in the rough setting of Ahlfors regular sets is built from the ground up and subsequently supports a Calderón-Zygmund theory on this brand of Morrey space in the optimal geometric environment of uniformly rectifiable sets. A thorough duality theory for such Morrey spaces is also developed and ushers in a never-before-seen Calderón-Zygmund theory for Muckenhoupt-weighted Block spaces. Both weighted Morrey and Block spaces are also considered through the lens of (generalized) Banach function spaces, and ultimately, a variety of boundary value problems are formulated and solved with boundary data arbitrarily prescribed from either scale of space. The fairly self-contained nature of this monograph ensures that graduate students, researchers, and professionals in a variety of fields, e.g., function space theory, harmonic analysis, and PDE, will find this monograph a welcome and valuable addition to the mathematical literature.

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate 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. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Traditionally, the label “Calderón-Zygmund theory” has been applied to a distinguished body of works primarily pertaining to the mapping properties of singular integral operators on Lebesgue spaces, in various geometric settings. Volume IV amounts to a versatile Calderón-Zygmund theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries, considered on a diverse range of function spaces. Novel applications to complex analysis in several variables are also explored here.

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate 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. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. The ultimate goal in Volume V is to prove well-posedness and Fredholm solvability results concerning boundary value problems for elliptic second-order homogeneous constant (complex) coefficient systems, and domains of a rather general geometric nature. The formulation of the boundary value problems treated here is optimal from a multitude of points of view, having to do with geometry, functional analysis (through the consideration of a large variety of scales of function spaces), topology, and partial differential equations.

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.

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate 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. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.