Geometric harmonic analysis IV. Boundary layer potentials in uniformly rectifiable domains, and applications to complex analysis (Q6162159)

The present book is the fourth in a series of five volumes, at the confluence of Harmonic Analysis, Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles: Volume~I: A Sharp Divergence Theorem with Nontangential Pointwise Traces; Volume~II: Function Spaces Measuring Size and Smoothness on Rough Sets; Volume~III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering; Volume~IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis; Volume~V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems. The main objective of the series is to produce tools that can treat efficiently boundary value problems for elliptic systems in inclusive geometric settings, beyond the category of Lipschitz domains. In this fourth volume, the bulk of the results amounts to a versatile Calderón-Zygmund theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries. The picture that emerges is that Calderón-Zygmund theory is a multi-faceted body of results aimed at describing how singular integral operators behave in many geometric and analytic settings. Applications to Complex Analysis in several variables are subsequently presented, starting from the realization that many natural integral operators in this setting, such as the Bochner-Martinelli operator, are actual particular cases of double-layer potential operators associated with the complex Laplacian. What follows is a concise description of the contents of each chapter. Chapter 1 focuses on singular integral operators (SIOs) of boundary layer type on Lebesgue and Sobolev spaces. Generic Calderón-Zygmund convolution-type SIOs [Volume~III, Chapter~2] are not expected to induce well-defined mappings on Sobolev spaces on uniformly rectifiable (UR) sets, as this requires a special algebraic structure of their integral kernels. Topics treated in this chapter include the history and physical interpretations of the classical harmonic layer potentials, ``tangential'' singular integral operators, volume and integral operators of boundary layer type associated with a given open set of locally finite perimeter and a given weakly elliptic system, a multitude of relevant examples and alternative points of view, a rich function theory of Calderón-Zygmund type for boundary layer potentials associated with a given weakly elliptic system and an open set with a uniformly rectifiable boundary, the interpretation of the Cauchy and Cauchy-Clifford operators as double-layer potential operators, and how to modify boundary layer potential operators to increase the class of functions to which they may be applied. Chapter 2 concentrates on layer potential operators acting on Hardy, BMO, VMO, and Hölder spaces defined on boundaries of UR domains. A fundamental aspect of this analysis is that a special algebraic structure is required of the integral kernel for a singular integral operator to map either of these spaces into itself and the brand of Divergence Theorem produced in Volume I plays a significant part. In fact, the same type of philosophy prevails in relation to the action of double-layer potential operators on Calderón, Morrey-Campanato, and Morrey spaces discussed in Chapter 3, and also for the action of double layer potential operators on Besov and Triebel-Lizorkin spaces, treated in Chapter 4. Chapter 5 addresses the following basic question: describe the most general classes of singular integral operators on the boundary of an arbitrary given UR domain $\Omega\subset{\mathbb{R}}^n$ which map Hardy, BMO, VMO, Hölder, Besov, and Triebel-Lizorkin spaces defined on $\partial\Omega$ boundedly into themselves. The authors provide an answer through the introduction of what they call generalized double-layer operators. They also take a look at Riesz transforms from the point of view of generalized double layers. In Chapter 6 the authors develop a theory of boundary layer potentials associated with the Stokes system of linear hydrostatics, and related topics. Among other things, they establish Green-type formulas, derive mapping properties for the aforementioned boundary layer potential operators, and prove Fatou-type results, in settings that are sharp from a geometric/analytic point of view. Once again, the brand of Divergence Theorem discussed in Volume~I plays a prominent role in carrying out this program. Chapter 7 contains a multitude of applications of the body of results developed so far in the area of Geometric Harmonic Analysis to the field of Complex Analysis of Several Variables. As is well known, Complex Analysis, Geometric Measure Theory and Harmonic Analysis interface tightly in the complex plane. However, this rich interplay between these branches of mathematics has been considerably less explored in the higher-dimensional setting, involving several complex variables. The main goal of the current chapter is to further the present understanding of this aspect. Themes covered include CR-functions and differential forms on boundaries of sets of locally finite perimeter, integration by parts formulas involving the \(d\)-bar operator on sets of locally finite perimeter, the Bochner-Martinelli integral operator, a sharp version of the Bochner-Martinelli-Koppelman formula, the Extension Problem for CR-functions in a variety of spaces on boundaries of Ahlfors regular (and UR) domains. Chapter 8 focuses on the study of Hardy spaces for certain second-order weakly elliptic operators in the complex plane, such as the Bitsadze operator in the unit disk. The purpose of the chapter is to characterize the space of null solutions and to identify precisely the corresponding spaces of boundary traces.