Pointwise estimates for a class of singular integrals and higher commutators

Qian Tao; Li Chun

doi:10.1007/bf02582027

Pointwise estimates for commutator singular integrals

Studia Mathematica ◽

10.4064/sm-62-1-1-15 ◽

1978 ◽

Vol 62 (1) ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

B. Baishanski ◽

R. Coifman

Keyword(s):

Singular Integrals ◽

Pointwise Estimates

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Weighted Estimates for Maximal Commutators of Multilinear Singular Integrals

Journal of Function Spaces and Applications ◽

10.1155/2012/128520 ◽

2012 ◽

Vol 2012 ◽

pp. 1-20 ◽

Cited By ~ 1

Author(s):

Dongxiang Chen ◽

Suzhen Mao

Keyword(s):

Linear Operator ◽

Lipschitz Function ◽

Singular Integrals ◽

Pointwise Estimates ◽

Multilinear Commutator ◽

Weighted Lipschitz Function ◽

Weighted Estimates ◽

Sharp Function ◽

Multilinear Commutators ◽

Multilinear Singular Integrals

This paper is concerned with the pointwise estimates for the sharp function of the maximal multilinear commutatorsTΣb*and maximal iterated commutatorTΠb*, generalized bym-linear operatorTand a weighted Lipschitz functionb. The(Lp1(μ)×⋯×Lpm(μ),Lr(μ1-r))boundedness and the(Lp1(μ)×⋯×Lpm(μ),Lr(μ1-mr))boundedness are obtained for maximal multilinear commutatorTΣb*and maximal iterated commutatorTΠb*, respectively.

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Pointwise estimates of one-sided approximations of one class of singular integrals

Researches in Mathematics ◽

10.15421/241317 ◽

2013 ◽

Vol 21 ◽

pp. 141

Author(s):

A.M. Pas'ko ◽

O.O. Kolesnyk

Keyword(s):

Singular Integrals ◽

Algebraic Polynomials ◽

Pointwise Estimates ◽

Pointwise Estimation

The pointwise estimation of the approximation to the class ${\breve{W}}_{\infty}^r$, $r \geqslant 1$, by algebraic polynomials is established.

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Multi-parameter Singular Integrals II: General Theory

10.23943/princeton/9780691162515.003.0005 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

General Theory ◽

Singular Integral ◽

Singular Integrals ◽

Integral Operators ◽

Singular Integral Operators ◽

Main Issue

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

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The Calder´on-Zygmund Theory I: Ellipticity

10.23943/princeton/9780691162515.003.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Classical Theory ◽

Singular Integral ◽

Singular Integrals ◽

Differential Operators ◽

Integral Operators ◽

Singular Integral Operators ◽

Single Parameter ◽

Partial Differential ◽

Translation Invariant ◽

Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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Multi-parameter Singular Integrals. (AM-189)

10.23943/princeton/9780691162515.001.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Partial Differential Equations ◽

Graduate Students ◽

General Theory ◽

Differential Equations ◽

Singular Integrals ◽

Main Tool ◽

Classical Theorem ◽

Quantitative Version ◽

Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

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