Multi-parameter Carnot-Carath´eodory Geometry

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Singular Integral ◽  
Maximal Function ◽  
Integral Operators ◽  
Product Type ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Square Function ◽  
Frobenius Theorem ◽  
Parameter Theory

This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the results are simple variants of results in the single-parameter theory. When the balls are not of product type, these ideas become more difficult. What saves the day is the quantitative Frobenius theorem given in Chapter 2. This can be used to estimate certain integrals, as well as develop an appropriate maximal function and an appropriate Littlewood–Paley square function, all of which are essential to our study of singular integral operators.

The Calder´on-Zygmund Theory I: Ellipticity

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.

scholarly journals Boundedness of Singular Integrals on Hardy Type Spaces Associated with Schrödinger Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Jianfeng Dong ◽  
Jizheng Huang ◽  
Heping Liu
Keyword(s):  
Molecular Characterization ◽  
Singular Integral ◽  
Maximal Function ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Hardy Type ◽  
Type Spaces ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator onRn,n≥3, whereV≢0is a nonnegative potential belonging to the reverse Hölder classBn/2. The Hardy type spacesHLp, n/(n+δ) <p≤1,for someδ>0, are defined in terms of the maximal function with respect to the semigroup{e-tL}t>0. In this paper, we investigate the bounded properties of some singular integral operators related toL, such asLiγand∇L-1/2, on spacesHLp. We give the molecular characterization ofHLp, which is used to establish theHLp-boundedness of singular integrals.

scholarly journals $M^2$-type sharp estimates and weighted boundedness for commutators related to singular integral operators satisfying a variant of Hörmander's condition

2013 ◽  
Vol 31 (2) ◽  
pp. 129
Author(s):  
Liu Lanzheliu
Keyword(s):  
Singular Integral ◽  
Maximal Function ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Sharp Maximal Function ◽  
Sharp Estimates ◽  
Hörmander’S Condition

In this paper, we prove the $M^k$-type sharp maximal function estimates for the commutators related to some singular integral operators satisfying a variant of Hörmander's condition.  As an application, we obtain the weighted boundedness of the commutators on Lebesgue and Morrey spaces.

The Calder´on-Zygmund Theory II: Maximal Hypoellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Differential Geometry ◽  
Singular Integral ◽  
Integral Operators ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Singular Integral Operators ◽  
Frobenius Theorem ◽  
Quantitative Version ◽  
Parameter Case

This chapter remains in the single-parameter case and turns to the case when the metric is a Carnot–Carathéodory (or sub-Riemannian) metric. It defines a class of singular integral operators adapted to this metric. The chapter has two major themes. The first is a more general reprise of the trichotomy described in Chapter 1 (Theorem 2.0.29). The second theme is a generalization of the fact that Euclidean singular integral operators are closely related to elliptic partial differential equations. The chapter also introduces a quantitative version of the classical Frobenius theorem from differential geometry. This “quantitative Frobenius theorem” can be thought of as yielding “scaling maps” which are well adapted to the Carnot–Carathéodory geometry, and is of central use throughout the rest of the monograph.

Calderon-Zygmund theory for singular integral operators associated with second-order elliptic partial differential systems on rough subdomains of Riemannian manifolds

2016 ◽  
Author(s):  
◽  
Brock Allen Schmutzler
Keyword(s):  
Singular Integral ◽  
Riemannian Manifolds ◽  
Measure Theory ◽  
Elliptic Boundary ◽  
Integral Operators ◽  
Geometric Measure Theory ◽  
Singular Integral Operators ◽  
Square Function ◽  
University Of Missouri ◽  
Geometric Measure

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This dissertation is a treatise on the theory of Calderon-Zygmund type singular integral operators capable of handling boundary layer potentials arising naturally in the treatment of elliptic boundary value problems on rough subdomains of Riemannian manifolds, where the nature of the underlying domain is very general, and is best described in the language of geometric measure theory. The resulting theory is a blend of harmonic analysis, differential geometry, and geometric measure theory which includes results pertaining to nontangential boundedness and pointwise traces (jump formulas), square-function and Carleson measure estimates, as well as compactness criteria on Lebesgue and Sobolev spaces.

Multi-parameter Singular Integrals II: General Theory

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.

scholarly journals The Calderón–Zygmund Theorem with an $$L^1$$ Mean Hörmander Condition

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Soichiro Suzuki
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Acta Math ◽  
Limited Range ◽  
Singular Integral Operators ◽  
Convolution Type ◽  
Worst Case ◽  
Hörmander Condition

AbstractIn 2019, Grafakos and Stockdale introduced an $$L^q$$ L q mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the $$L^p$$ L p boundedness for the “limited-range” instead of $$1< p < \infty $$ 1 < p < ∞ . However, in this paper, we show that the $$L^q$$ L q mean Hörmander condition is actually enough to obtain the $$L^p$$ L p boundedness for all $$1< p < \infty $$ 1 < p < ∞ even in the worst case $$q=1$$ q = 1 . We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the $$L^2$$ L 2 boundedness for convolution type singular integral operators under the $$L^1$$ L 1 mean Hörmander condition.

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