Maximal Function Methods for Sobolev Spaces

| Inequalities (Mathematics) Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Maximal functions, Littlewood-Paley theory. | Functional analysis – Linear function spaces and their duals – Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems. | Real functions – Inequalities – Inequalities involving derivatives and differential and integral operators. | Measure and integration – Classical measure theory – Contents, measures, outer measures, capacities. | Potential theory – Higher-dimensional theory – Potentials and capacities, extremal length and related notions in higher dimensions. | Partial differential equations – General topics in partial differential equations – Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals. | Partial differential equations – Elliptic equations and systems – Quasilinear elliptic equations with p -Laplacian. | Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Harmonic analysis and This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy’s inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy’s inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy’s inequalities. At the end of the book, the theory of weak solutions to the p -Laplace equation and the use of maximal function techniques is this context are discussed. The book is to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.