In this book we suggest a unified method of constructing near-minimizers for certain important functionals arising in approximation, harmonic analysis and ill-posed problems and most widely used in interpolation theory. The constructions are based on far-reaching refinements of the classical CalderA³naZygmund decomposition. These new CalderA³naZygmund 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 near-minimizers) under the action of CalderA³naZygmund 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 self-contained presentation of the main topic. There we discuss the classical covering results mentioned above, various spectacular applications of the classical CalderA³naZygmund 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.Finite multiplicity (in fact, even finite overlap) follows because this occurs already for the family {(1 I eO)Qw, } (recall that Km, C (1 + e0)Q$, ). So, only the Whitney property needs a proof. iGI First, we show that U Qxt- : U(1 + 50)Qx4. 8.1. Whitney anbsp;...

Title | : | Extremal Problems in Interpolation Theory, Whitney-Besicovitch Coverings, and Singular Integrals |

Author | : | Sergey Kislyakov, Natan Kruglyak |

Publisher | : | Springer Science & Business Media - 2012-10-29 |

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