"This book discusses advances in maximal function methods related to Poincare 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 Holder 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 directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations"-- Provided by publisher

Includes bibliographical references (pages 317-333) and index

Maximal functions -- Lipschitz and Sobolev functions -- Sobolev and Poincare inequalities -- Pointwise inequalities for Sobolev functions -- Capacities and fine properties of Sobolev functions -- Hardy's inequalities -- Density conditions -- Muckenhoupt weights -- Weighted maximal and Poincareinequalities -- Distance weights and Hardy-Sobolev inequalities -- The p-Laplace equation -- Stability results for the p-Laplace equation