By Sergey Kislyakov, Natan Kruglyak

ISBN-10: 3034804687

ISBN-13: 9783034804684

During this publication we propose a unified approach to developing near-minimizers for yes vital functionals bobbing up in approximation, harmonic research and ill-posed difficulties and most generally utilized in interpolation thought. The buildings are in response to far-reaching refinements of the classical Calderón–Zygmund decomposition. those new Calderón–Zygmund decompositions in flip are produced with assistance from new masking theorems that mix many impressive gains of classical effects tested through Besicovitch, Whitney and Wiener. in lots of instances the minimizers built within the ebook are strong (i.e., stay near-minimizers) lower than the motion of Calderón–Zygmund singular fundamental operators. The publication is split into components. whereas the recent approach is gifted in nice aspect within the moment half, the 1st is especially dedicated to the must haves wanted for a self-contained presentation of the most subject. There we speak about the classical masking effects pointed out above, quite a few incredible functions of the classical Calderón–Zygmund decompositions, and the connection of all this to actual interpolation. It additionally serves as a brief creation to such very important themes as areas of gentle features or singular integrals.

Table of Contents

Cover

Extremal difficulties in Interpolation conception, Whitney-Besicovitch Coverings, and Singular Integrals

ISBN 9783034804684 e-ISBN 9783034804691

Contents

Preface

Introduction

Definitions, notation, and a few usual facts

Geometry

Spaces

Functionals of genuine interpolation

Theorems

wrong proofs

Part I Background

bankruptcy 1 Classical Calder�n-Zygmund decomposition and genuine interpolation

1.1 Riesz emerging solar lemma and the Calder�n-Zygmund procedure

o 1.1.1 Riesz emerging solar lemma

o 1.1.2 Calder�n-Zygmund lemma

o 1.1.3 Calder�n-Zygmund decomposition

o 1.1.4 A susceptible variety 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 dating with actual interpolation

1.4 An basic balance theorem

o 1.4.1 an evidence with a lot interpolation

o 1.4.2 Stabilization � los angeles Bourgain

o 1.4.3 a few consequences

Notes and remarks

bankruptcy 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 basic definition
o 2.2.1 Examples
o 2.2.2 extra information
2.3 Vector-valued analogs
Notes and remarks
bankruptcy three Classical overlaying theorems
3.1 Classical masking theorems and walls 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 walls of unity
3.2 one other Calder�n-Zygmund procedure
3.3 balance of near-minimizers for the couple (L1, L8)
o 3.3.1 assertion and proof
o 3.3.2 Vector type of the steadiness theorem
Notes and remarks
bankruptcy four areas of gentle features and operators on them
4.1 Summary
o 4.1.1 Homogeneous areas of tender functions
# Sobolev spaces
# Morrey-Campanato spaces
# moderate values of okay within the definition of Morrey-Campanato spaces
# Morrey spaces
# 0 smoothness
o 4.1.2 Singular critical operators
4.2 Morrey-Campanato areas: 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 with regards to wavelet expansions
o 4.5.1 extra common operators
o 4.5.2 Consequences
o 4.5.3 An passed over proof
4.6 susceptible L1-boundedness
Notes and remarks
bankruptcy five a few themes in interpolation
5.1 major 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 undemanding balance theorem revisited
5.6 K-closed subcouples and stability
5.7 Linearization
Notes and remarks
bankruptcy 6 Regularization for Banach spaces
Notes and remarks
bankruptcy 7 balance for analytic Hardy spaces
Notes and remarks
Part II complicated theory
bankruptcy eight managed coverings
8.1 Whitney lemma and a theorem approximately Lipschitz families
o 8.1.1 Auxiliary lemmas
o 8.1.2 Finite overlap
o 8.1.3 Meshing set of rules and the robust engagement lemma
o 8.1.4 transformed Besicovitch q-process
o 8.1.5 evidence of Theorem 8.16
o 8.1.6 facts of Theorem 8.9
o 8.1.7 evidence of Theorem 8.13
8.2 managed extension and renovation of the a-capacity
o 8.2.1 The Besicovitch approach with a Lipschitz condition
o 8.2.2 building of a WB-covering
o 8.2.3 evidence of the managed extension theorem
o 8.2.4 evidence of the concept at the renovation of a-capacity for aIE(1-1/n,1)
8.3 managed contraction and upkeep of the a-capacity
o 8.3.1 Besicovitch q-process with a Lipschitz for managed contraction
o 8.3.2 building of a WB-covering
o 8.3.3 evidence of the contraction theorem
8.4 renovation of the a-capacity (a < 0)
Notes and remarks
bankruptcy nine development of near-minimizers
9.1 Estimates for derivatives of approximants
9.2 Near-minimizers for Sobolev areas: the (Lp, Wkq)
o 9.2.1 Near-minimizers for the couple (Lp, Wkq)
o 9.2.2 Near-minimizers for the couple (Lp, Wkq) while q!=p
o 9.2.3 assertion and facts of the most result
9.3 Near-minimizers for Morrey-Campanato areas: the (Lp, Ca,k p)
o 9.3.1 set of rules for developing near-minimizers
o 9.3.2 assertion and the evidence of the most result
Notes and remarks
bankruptcy 10 balance of near-minimizers
10.1 building of approximating polynomials
10.2 balance theorems: statements and applications
o 10.2.1 Statements
o 10.2.2 Applications
# Shift in smoothness
# A estate of wavelet expansions
10.3 evidence of Theorems 10.4-10.6
o 10.3.1 facts of the most lemma
o Estimate of the second one and 3rd summands in (10.24)
o Estimate of the 1st summand in (10.24)
Notes and remarks
bankruptcy eleven The passed over case of a restrict exponent
11.1 Description of the algorithm
11.2 important effects, and descriptions of the proofs
o 11.2.1 assertion of the most results
11.3 Proofs
o 11.3.1 The case of t >= t*

o 11.3.2 Lemmas legitimate within 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 final algorithm

A.2 Near-minimizers for Morrey areas equipped at the foundation of Brudnyi spaces

A.2.1 Auxiliary lemmas

A.2.2 facts of the most end result (Theorem A.3)

A.3 Near-minimizers for Morrey areas equipped at the foundation of Triebel-Lizorkin spaces

A.3.1 Auxiliary lemmas

A.3.2 evidence of the most outcome (Theorem A.12)

Notes and remarks

Bibliography

Index