Title
Extremal Problems in Interpolation Theory, WhitneyBesicovitch Coverings, and Singular Integrals (Monografie Matematyczne, 74),Used
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In this book we suggest a unified method of constructing nearminimizers for certain important functionals arising in approximation, harmonic analysis and illposed problems and most widely used in interpolation theory. The constructions are based on farreaching refinements of the classical CaldernZygmund decomposition. These new CaldernZygmund decompositions in turn are produced with the help of new covering theorems that combine many remarkable features of classical results established by Besicovitch, Whitney and Wiener. In many cases the minimizers constructed in the book are stable (i.e., remain nearminimizers) under the action of CaldernZygmund singular integral operators.The book is divided into two parts. While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a selfcontained presentation of the main topic. There we discuss the classical covering results mentioned above, various spectacular applications of the classical CaldernZygmund decompositions, and the relationship of all this to real interpolation. It also serves as a quick introduction to such important topics as spaces of smooth functions or singular integrals.
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