Sobolev Spaces

The Sobolev spaces, i. e. the classes of functions with derivatives in L , occupy p an outstanding place in analysis. During the last two decades a substantial contribution to the study of these spaces has been made; so now solutions to many important problems connected with them are known. In the present monograph we consider various aspects of Sobolev space theory. Attention is paid mainly to the so called imbedding theorems. Such theorems, originally established by S. L. Sobolev in the 1930s, proved to be a useful tool in functional analysis and in the theory of linear and nonlinear par­ tial differential equations. We list some questions considered in this book. 1. What are the requirements on the measure f1, for the inequality q Front Matter....Pages I-XIX Introduction....Pages 1-5 Basic Properties of Sobolev Spaces....Pages 6-87 Inequalities for Gradients of Functions that Vanish on the Boundary....Pages 88-159 On Summability of Functions in the Space L 1 1 ( Ω )....Pages 160-190 On Summability of Functions in the Space L p 1 ( Ω )....Pages 191-269 On Continuity and Boundedness of Functions in Sobolev Spaces....Pages 270-295 On Functions in the Space B V(Ω) ....Pages 296-341 Certain Function Spaces, Capacities and Potentials....Pages 342-359 On Summability with Respect to an Arbitrary Measure of Functions with Fractional Derivatives....Pages 360-389 A Variant of Capacity....Pages 390-401 An Integral Inequality for Functions on a Cube....Pages 402-423 Imbedding of the Space into Other Function Spaces....Pages 424-452 The Imbedding ....Pages 453-468 Back Matter....Pages 469-488 The Sobolev spaces, i. e. the classes of functions with derivatives in L, occupy p an outstanding place in analysis. During the last two decades a substantial contribution to the study of these spaces has been made; so now solutions to many important problems connected with them are known. In the present monograph we consider various aspects of Sobolev space theory. Attention is paid mainly to the so called imbedding theorems. Such theorems, originally established by S.L. Sobolev in the 1930s, proved to be a useful tool in functional analysis and in the theory of linear and nonlinear parƯ tial differential equations. We list some questions considered in this book. 1. What are the requirements on the measure f1, for the inequality q