This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems – as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis – will find this text to be a valuable addition to the mathematical literature. Preface 6 Acknowledgments 7 Contents 8 1 Introduction 10 2 Geometric Measure Theory 36 2.1 Classes of Euclidean Sets of Locally Finite Perimeter 37 2.2 Reifenberg Flat Domains 62 2.3 Chord-Arc Curves in the Plane 75 2.4 The Class of Delta-Flat Ahlfors Regular Domains 94 2.5 The Decomposition Theorem 109 2.6 Chord-Arc Domains in the Plane 129 2.7 Dyadic Grids and Muckenhoupt Weights on Ahlfors Regular Sets 137 2.8 Sobolev Spaces on Ahlfors Regular Sets 154 3 Calderón–Zygmund Theory for Boundary Layers in UR Domains 171 3.1 Boundary Layer Potentials: The Setup 171 3.2 SIOs on Muckenhoupt Weighted Lebesgue and Sobolev Spaces 187 3.3 Distinguished Coefficient Tensors 208 4 Boundedness and Invertibility of Layer Potential Operators 249 4.1 Estimates for Euclidean Singular Integral Operators 249 4.2 Estimates for Certain Classes of Singular Integrals on UR Sets 267 4.3 Norm Estimates and Invertibility Results for Double Layers 302 4.4 Invertibility on Muckenhoupt Weighted Homogeneous Sobolev Spaces 326 4.5 Another Look at Double Layers for the Two-Dimensional Lamé System 338 5 Controlling the BMO Semi-Norm of the Unit Normal 346 5.1 Clifford Algebras and Cauchy–Clifford Operators 347 5.2 Estimating the BMO Semi-Norm of the Unit Normal 351 5.3 Using Riesz Transforms to Quantify Flatness 359 5.4 Using Riesz Transforms to Characterize Muckenhoupt Weights 362 6 Boundary Value Problems in Muckenhoupt Weighted Spaces 372 6.1 The Dirichlet Problem in Weighted Lebesgue Spaces 374 6.2 The Regularity Problem in Weighted Sobolev Spaces 386 6.3 The Neumann Problem in Weighted Lebesgue Spaces 403 6.4 The Transmission Problem in Weighted Lebesgue Spaces 418 7 Singular Integrals and Boundary Problems in Morrey and Block Spaces 439 7.1 Boundary Layer Potentials on Morrey and Block Spaces 439 7.2 Inverting Double Layer Operators on Morrey and Block Spaces 466 7.3 Invertibility on Morrey/Block-Based HomogeneousSobolev Spaces 473 7.4 Characterizing Flatness in Terms of Morrey and Block Spaces 481 7.5 Boundary Value Problems in Morrey and Block Spaces 487 8 Singular Integrals and Boundary Problems in Weighted Banach Function Spaces 503 8.1 Basic Properties and Extrapolation in Banach Function Spaces 503 8.2 Boundary Layer Potentials on Weighted Banach Function Spaces 517 8.3 Inverting Double Layer Operators on Weighted Banach Function Spaces 535 8.4 Invertibility on Homogeneous Weighted Banach Function-Based Sobolev Spaces 539 8.5 Characterizing Flatness in Terms of Weighted Banach Functions Spaces 546 8.6 Boundary Value Problems in Weighted Banach Function Spaces 552 8.7 Examples of Weighted Banach Function Spaces 570 8.7.1 Unweighted Banach Function Spaces 570 8.7.2 Rearrangement Invariant Banach Function Spaces 572 References 593 Subject Index 600 Symbol Index 603
