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

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 Calderón–Zygmund decomposition. These new Calderón–Zygmund 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 Calderón–Zygmund 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 Calderón–Zygmund 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. Table of Contents Cover Extremal Problems in Interpolation Theory, Whitney-Besicovitch Coverings, and Singular Integrals ISBN 9783034804684 e-ISBN 9783034804691 Contents Preface Introduction Definitions, notation, and some standard facts Geometry Spaces Functionals of real interpolation Theorems Two improper proofs Part I Background Chapter 1 Classical Calder�n-Zygmund decomposition and real interpolation 1.1 Riesz rising sun lemma and the Calder�n-Zygmund procedure o 1.1.1 Riesz rising sun lemma o 1.1.2 Calder�n-Zygmund lemma o 1.1.3 Calder�n-Zygmund decomposition o 1.1.4 A weak type inequality for linear operators o 1.1.5 Hardy-Littlewood maximal operator 1.2 Norms on BMO and Lipschitz spaces o 1.2.1 John-Nirenberg inequality o 1.2.2 Equivalence of Campanato norms 1.3 Relationship with real interpolation 1.4 An elementary stability theorem o 1.4.1 A proof with much interpolation o 1.4.2 Stabilization � la Bourgain o 1.4.3 Some consequences Notes and remarks Chapter 2 Singular integrals 2.1 Hilbert transformation o 2.1.1 Hilbert transformation on L1 o 2.1.2 The operator H on Lp, 1 < p < infty 2.2 General definition o 2.2.1 Examples o 2.2.2 Additional information 2.3 Vector-valued analogs Notes and remarks Chapter 3 Classical covering theorems 3.1 Classical covering theorems and partitions of unity o 3.1.1 The Besicovitch q-process o 3.1.2 Besicovitch theorem o 3.1.3 Wiener lemma o 3.1.4 Whitney lemma, WB-coverings, and partitions of unity 3.2 Another Calder�n-Zygmund procedure 3.3 Stability of near-minimizers for the couple (L1, L8) o 3.3.1 Statement and proof o 3.3.2 Vector form of the stability theorem Notes and remarks Chapter 4 Spaces of smooth functions and operators on them 4.1 Summary o 4.1.1 Homogeneous spaces of smooth functions # Sobolev spaces # Morrey-Campanato spaces # Reasonable values of k in the definition of Morrey-Campanato spaces # Morrey spaces # Zero smoothness o 4.1.2 Singular integral operators 4.2 Morrey-Campanato spaces: proofs 4.3 BMO and atomic H1 4.4 Continuity of operators on BMO and Lipschitz spaces o 4.4.1 A pointwise estimate o 4.4.2 Norm estimates 4.5 Singular integrals related to wavelet expansions o 4.5.1 More general operators o 4.5.2 Consequences o 4.5.3 An omitted proof 4.6 Weak L1-boundedness Notes and remarks Chapter 5 Some topics in interpolation 5.1 Main notions 5.2 Near-minimizers and interpolation 5.3 Near-minimizers for Lp,q- and K-functionals 5.4 Near-minimizers for E- and K-functionals 5.5 The elementary stability theorem revisited 5.6 K-closed subcouples and stability 5.7 Linearization Notes and remarks Chapter 6 Regularization for Banach spaces Notes and remarks Chapter 7 Stability for analytic Hardy spaces Notes and remarks Part II Advanced theory Chapter 8 Controlled coverings 8.1 Whitney lemma and a theorem about Lipschitz families o 8.1.1 Auxiliary lemmas o 8.1.2 Finite overlap o 8.1.3 Meshing algorithm and the strong engagement lemma o 8.1.4 Modified Besicovitch q-process o 8.1.5 Proof of Theorem 8.16 o 8.1.6 Proof of Theorem 8.9 o 8.1.7 Proof of Theorem 8.13 8.2 Controlled extension and preservation of the a-capacity o 8.2.1 The Besicovitch process with a Lipschitz condition o 8.2.2 Construction of a WB-covering o 8.2.3 Proof of the controlled extension theorem o 8.2.4 Proof of the theorem on the preservation of a-capacity for aIE(1-1/n,1) 8.3 Controlled contraction and preservation of the a-capacity o 8.3.1 Besicovitch q-process with a Lipschitz condition for controlled contraction o 8.3.2 Construction of a WB-covering o 8.3.3 Proof of the contraction theorem 8.4 Preservation of the a-capacity (a < 0) Notes and remarks Chapter 9 Construction of near-minimizers 9.1 Estimates for derivatives of approximants 9.2 Near-minimizers for Sobolev spaces: the couples (Lp, Wkq) o 9.2.1 Near-minimizers for the couple (Lp, Wkq) o 9.2.2 Near-minimizers for the couple (Lp, Wkq) when q!=p o 9.2.3 Statement and proof of the main result 9.3 Near-minimizers for Morrey-Campanato spaces: the couples (Lp, Ca,k p) o 9.3.1 Algorithm for constructing near-minimizers o 9.3.2 Statement and the proof of the main result Notes and remarks Chapter 10 Stability of near-minimizers 10.1 Construction of approximating polynomials 10.2 Stability theorems: statements and applications o 10.2.1 Statements o 10.2.2 Applications # Shift in smoothness # A property of wavelet expansions 10.3 Proof of Theorems 10.4-10.6 o 10.3.1 Proof of the main lemma o Estimate of the second and third summands in (10.24) o Estimate of the first summand in (10.24) Notes and remarks Chapter 11 The omitted case of a limit exponent 11.1 Description of the algorithm 11.2 Principal results, and outlines of the proofs o 11.2.1 Statement of the main results 11.3 Proofs o 11.3.1 The case of t >= t* o 11.3.2 Lemmas valid in the multidimensional case and Theorem 11.3 o 11.3.3 Geometric lemmas and the proofs of Theorems 11.4 and 11.5 Notes and remarks Chapter A Appendix. Near-minimizers for Brudnyi and Triebel-Lizorkin spaces A.1 Description of the general algorithm A.2 Near-minimizers for Morrey spaces built on the basis of Brudnyi spaces A.2.1 Auxiliary lemmas A.2.2 Proof of the main result (Theorem A.3) A.3 Near-minimizers for Morrey spaces built on the basis of Triebel-Lizorkin spaces A.3.1 Auxiliary lemmas A.3.2 Proof of the main result (Theorem A.12) Notes and remarks Bibliography Index