This volume is dedicated to the centenary of the outstanding mathematician of the XXth century Sergey Sobolev and, in a sense, to his celebrated work On a theorem of functional analysis published in 1938, exactly 70 years ago, where the original Sobolev inequality was proved. This double event is a good case to gather experts for presenting the latest results on the study of Sobolev inequalities which play a fundamental role in analysis, the theory of partial differential equations, mathematical physics, and differential geometry. In particular, the following topics are discussed: Sobolev type inequalities on manifolds and metric measure spaces, traces, inequalities with weights, unfamiliar settings of Sobolev type inequalities, Sobolev mappings between manifolds and vector spaces, properties of maximal functions in Sobolev spaces, the sharpness of constants in inequalities, etc. The volume opens with a nice survey reminiscence My Love Affair with the Sobolev Inequality by David R. Adams.

Title is also available as part of a set: Sobolev Spaces in Mathematics I, II, III (978-0-387-85791-6)

My Love Affair with the Sobolev Inequality.- Maximal Functions in Sobolev Spaces.- Hardy Type Inequalities via Riccati and Sturm-Liouville Equations.- Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles.- Inequalities of Hardy-Sobolev Type in Carnot-Carathéodory Spaces.- Sobolev Embeddings and Hardy Operators.- Sobolev Mappings between Manifolds and Metric Spaces.- A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions.- Optimality of Function Spaces in Sobolev Embeddings.- On the Hardy-Sobolev-Maz'ya Inequality and Its Generalizations.- Sobolev Inequalities in Familiar and Unfamiliar Settings.- A Universality Property of Sobolev Spaces in Metric Measure Spaces.- Cocompact Imbeddings and Structure of Weakly Convergent Sequences.