scholarly journals The common Hardy space and BMO space for singular integral operators associated with isotropic and anisotropic homogeneity

2014 ◽  
Vol 414 (1) ◽  
pp. 480-487
Author(s):  
Chaoqiang Tan ◽  
Xiaosheng Zhuang
Keyword(s):  
Hardy Space ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Bmo Space ◽  
The Common

The Hardy Space H1 on Manifolds and Submanifolds

1972 ◽  
Vol 24 (5) ◽  
pp. 915-925 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Partial Differential Equations ◽  
Hardy Space ◽  
Euclidean Space ◽  
Differential Equations ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Riesz Transforms ◽  
Partial Differential ◽  
Integrable Functions

It is well-known that the space L1(Rn) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H1(Rn) of integrable functions whose Riesz transforms are integrable.

scholarly journals Herz-Type Hardy Spaces Associated with Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Yan Chai ◽  
Yaoyao Han ◽  
Kai Zhao
Keyword(s):  
Differential Operator ◽  
Hardy Space ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Integral Operators ◽  
Singular Integral Operators

Suppose L is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator L. Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator L. As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.

scholarly journals Brown-Halmos Type Theorems of Weighted Toeplitz Operators

1998 ◽  
Vol 41 (2) ◽  
pp. 196-206 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Hardy Space ◽  
Singular Integral ◽  
Lebesgue Space ◽  
Toeplitz Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Weighted Hardy Space

AbstractThe spectra of the Toeplitz operators on the weighted Hardy space H2(Wdθ / 2π) and the Hardy space Hp(dθ / 2π), and the singular integral operators on the Lebesgue space L2(dθ / 2π) are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

Singular Integral Operators and Essential Commutativity on the Sphere

2010 ◽  
Vol 62 (4) ◽  
pp. 889-913 ◽  
Author(s):  
Jingbo Xia
Keyword(s):  
Hardy Space ◽  
Singular Integral ◽  
Unit Sphere ◽  
Toeplitz Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Essential Commutant

AbstractLet 𝒯 be the C*-algebra generated by the Toeplitz operators {T𝜑 : 𝜑 Є L∞(S, dσ)} on the Hardy space H2(S) of the unit sphere in Cn. It is well known that 𝒯 is contained in the essential commutant of {T𝜑 : 𝜑 Є VMO∩L∞(S, dσ)}. We show that the essential commutant of {T𝜑 : 𝜑 Є VMO∩L∞(S, dσ)} is strictly larger than 𝒯.

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.

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.

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