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.

Presents many significant new results on the topic of singular integral operators and boundary value problems Contributes to ongoing efforts to establish a bridge between analysis and geometry Includes many complete proofs appearing in publication for the first time

Autorentext

Juan José Marín is a harmonic analyst whose research interests also include boundary value problems and geometric measure theory. He received a Ph.D. in mathematics in 2019 from Universidad Aut'onoma de Madrid and Instituto de Ciencias Matem'aticas, Spain, working under the supervision of José María Martell and Marius Mitrea. José María Martell is a mathematician specializing in the areas of harmonic analysis, partial differential equations, and geometric measure theory. He received a Ph.D. in mathematics from Universidad Autónoma de Madrid, Spain, working under the supervision of José Garcia-Cuerva. José María Martell is currently serving as the director of Instituto de Matemáticas, Spain. Dorina Mitrea is a mathematician specializing in the areas of harmonic analysis, partial differential equations, geometric measure theory, and global analysis. She received a Ph.D. in mathematics from the University of Minnesota, working under the supervision of Eugene Fabes. Dorina Mitrea is currently serving as the chair of the Department of Mathematics, Baylor University, USA. Irina Mitrea is an L.H. Carnell Professor and chair of the Department of Mathematics at Temple University whose expertise lies at the interface between the areas of harmonic analysis, partial differential equations, and geometric measure theory. She received her Ph.D. in mathematics from the University of Minnesota, working under the supervision of Carlos Kenig and Mikhail Safanov. Irina Mitrea is a Fellow of the American Mathematical Society and a Fellow of the Association for Women in Mathematics. She received a Simons Foundation Fellowship, a Von Neumann Fellowship at the Institute for Advanced Study, Princeton, and is a recipient of the Ruth Michler Memorial Prize from the Association for Women in Mathematics. Marius Mitrea is a mathematician whose research interests lay at the confluence between harmonic analysis, partial differential equations, geometric measure theory, global analysis, and scattering. He received a Ph.D. in mathematics from the University of South Carolina, USA, working under the supervision of Björn D. Jawerth. Marius Mitrea is a Fellow of the American Mathematical Society.

Inhalt
Introduction.- Geometric Measure Theory.- Calderon-Zygmund Theory for Boundary Layers in UR Domains.- Boundedness and Invertibility of Layer Potential Operators.- Controlling the BMO Semi-Norm of the Unit Normal.- Boundary Value Problems in Muckenhoupt Weighted Spaces.- Singular Integrals and Boundary Problems in Morrey and Block Spaces.- Singular Integrals and Boundary Problems in Weighted Banach Function Spaces.