Certain Operators with Rough Singular Kernels

2003 ◽  
Vol 55 (3) ◽  
pp. 504-532 ◽  
Author(s):  
Jiecheng Chen ◽  
Dashan Fan ◽  
Yiming Ying
Keyword(s):  
Hardy Space ◽  
Sobolev Space ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Integrable Function ◽  
Bounded Function ◽  
Bounded Operator ◽  
Integral Operators ◽  
Test Functions ◽  
Singular Kernels

AbstractWe study the singular integral operatordefined on all test functions f, where b is a bounded function, α ≥ 0, Ω (yʻ) is an integrable function on the unit sphere Sn-1 satisfying certain cancellation conditions. We prove that, for 1 < p < ∞, TΩ,α extends to a bounded operator from the Sobolev space to the Lebesgue space Lp with Ω being a distribution in the Hardy space Hq(Sn-1) where . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for TΩ,α on the Hardy spaces, as well as the boundedness for the truncated maximal operator .

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.

Oscillatory singular integral operators with Hölder class kernels on Hardy spaces

2019 ◽  
Vol 31 (2) ◽  
pp. 535-542
Author(s):  
Yibiao Pan
Keyword(s):  
Singular Integral ◽  
Hardy Spaces ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Oscillatory Singular Integral ◽  
Hölder Class ◽  
Holder Class ◽  
Oscillatory Singular Integrals

AbstractA sharp logarithmic bound is established for the {H^{1}}-norm of oscillatory singular integrals with quadratic phases and Hölder class kernels. Prior results had relied on a {C^{1}}-assumption on the kernel.

scholarly journals Boundedness for a Class of Singular Integral Operators on Both Classical and Product Hardy Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Chaoqiang Tan
Keyword(s):  
Singular Integral ◽  
Hardy Spaces ◽  
General Class ◽  
Integral Operators ◽  
Singular Integral Operators

We found that the classical Calderón-Zygmund singular integral operators are bounded on both the classical Hardy spaces and the product Hardy spaces. The purpose of this paper is to extend this result to a more general class. More precisely, we introduce a class of singular integral operators including the classical Calderón-Zygmund singular integral operators and show that they are bounded on both the classical Hardy spaces and the product Hardy spaces.

BOUNDEDNESS OF SEVERAL INTEGRAL OPERATORS WITH BOUNDED VARIABLE KERNELS ON HARDY AND WEAK HARDY SPACES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350095 ◽  
Author(s):  
HUA WANG
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Variable Kernel ◽  
Marcinkiewicz Integrals ◽  
Decomposition Theory ◽  
Logarithmic Type ◽  
Lipschitz Conditions

In this paper, by using the atomic decomposition theory of Hardy space H1(ℝn) and weak Hardy space WH1(ℝn), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

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 The Boundedness on Mixed Hardy Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wei Ding ◽  
Meidi Qin ◽  
Yueping Zhu
Keyword(s):  
Singular Integral ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Direct Method ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
A Direct Method

The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journé class on mixed Hardy spaces by a direct method.

scholarly journals Hardy spaces estimates for multilinear operators with homogeneous kernels

2003 ◽  
Vol 170 ◽  
pp. 117-133 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Fractional Integral ◽  
Integral Operators ◽  
Kernel Functions ◽  
Multilinear Operators ◽  
Product Spaces ◽  
Bounded Operators ◽  
Fractional Integral Operators ◽  
Bmo Function

AbstractIn this paper the authors prove that a class of multilinear operators formed by the singular integral or fractional integral operators with homogeneous kernels are bounded operators from the product spaces Lp1 × Lp2 × · · · × LpK (ℝn) to the Hardy spaces Hq (ℝn) and the weak Hardy space Hq,∞(ℝn), where the kernel functions Ωij satisfy only the Ls-Dini conditions. As an application of this result, we obtain the (Lp, Lq) boundedness for a class of commutator of the fractional integral with homogeneous kernels and BMO function.

scholarly journals On an extension of singular integrals along manifolds of finite type

2006 ◽  
Vol 2006 ◽  
pp. 1-16
Author(s):  
Abdelnaser Al-Hasan ◽  
Dashan Fan
Keyword(s):  
Sobolev Space ◽  
Finite Type ◽  
Singular Integral ◽  
Lebesgue Space ◽  
Compact Manifold ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Homogeneous Sobolev Space

We extend theLp-boundedness of a class of singular integral operators under theH1kernel condition on a compact manifold from the homogeneous Sobolev spaceL˙αp(ℝn)to the Lebesgue spaceLp(ℝn).

scholarly journals Rough singular integrals on product spaces

2004 ◽  
Vol 2004 (67) ◽  
pp. 3671-3684 ◽  
Author(s):  
Ahmad Al-Salman ◽  
Hussain Al-Qassem
Keyword(s):  
Finite Type ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Product Spaces ◽  
Lebesgue Spaces ◽  
Block Spaces ◽  
Singular Kernels

We study the mapping properties of singular integral operators defined by mappings of finite type. We prove that such singular integral operators are bounded on the Lebesgue spaces under the condition that the singular kernels are allowed to be in certain block spaces.

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