This book develops a theory of extrapolation spaces with applications to classical and modern analysis. Extrapolation theory aims to provide a general framework to study limiting estimates in analysis. The book also considers the role that optimal decompositions play in limiting inequalities incl. commutator estimates. Most of the results presented are new or have not appeared in book form before. A special feature of the book are the applications to other areas of analysis. Among them Sobolev imbedding theorems in different contexts including logarithmic Sobolev inequalities are obtained, commutator estimates are connected to the theory of comp. compactness, a connection with maximal regularity for abstract parabolic equations is shown, sharp estimates for maximal operators in classical Fourier analysis are derived. Background on extrapolation theory.- K/J inequalities and limiting embedding theorems.- Calculations with the ? method and applications.- Bilinear extrapolation and a limiting case of a theorem by Cwikel.- Extrapolation, reiteration, and applications.- Estimates for commutators in real interpolation.- Sobolev imbedding theorems and extrapolation of infinitely many operators.- Some remarks on extrapolation spaces and abstract parabolic equations.- Optimal decompositions, scales, and Nash-Moser iteration.
