On a measure of non-compactness for singular integrals

It is proved that there exists no weight pair(v, w)for which a singular integral operator is compact from the weighted Lebesgue spaceLwp(Rn)toLvp(Rn). Moreover, a measure of non-compatness for this operator is estimated from below. Analogous problems for Cauchy singular integrals defined on Jordan smooth curves are studied.

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Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

Journal of Function Spaces ◽

10.1155/2017/3764142 ◽

2017 ◽

Vol 2017 ◽

pp. 1-9

Author(s):

Wei Wang ◽

Jingshi Xu

Keyword(s):

Integral Operator ◽

Singular Integral Operator ◽

Singular Integral ◽

Singular Integrals ◽

Sufficient Conditions ◽

Morrey Spaces ◽

Bmo Function

We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

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Compactness of Commutators for Singular Integrals on Morrey Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-043-1 ◽

2012 ◽

Vol 64 (2) ◽

pp. 257-281 ◽

Cited By ~ 32

Author(s):

Yanping Chen ◽

Yong Ding ◽

Xinxia Wang

Keyword(s):

Integral Operator ◽

Compact Operator ◽

Singular Integral Operator ◽

Singular Integral ◽

Morrey Space ◽

Singular Integrals ◽

Morrey Spaces ◽

Compact Set ◽

Sufficient Condition

AbstractIn this paper we characterize the compactness of the commutator [b, T] for the singular integral operator on the Morrey spaces . More precisely, we prove that if , the -closure of , then [b, T] is a compact operator on the Morrey spaces for ∞ < p < ∞ and 0 < ⋋ < n. Conversely, if and [b, T] is a compact operator on the for some p (1 < p < ∞), then . Moreover, the boundedness of a rough singular integral operator T and its commutator [b, T] on are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

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Weighted norm inequalities for a class of rough singular integrals

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.657 ◽

2005 ◽

Vol 2005 (5) ◽

pp. 657-669

Author(s):

H. M. Al-Qassem

Keyword(s):

Integral Operator ◽

Singular Integral Operator ◽

Singular Integral ◽

Singular Integrals ◽

Singular Operator ◽

Weighted Norm ◽

Weighted Norm Inequalities ◽

Norm Inequalities

Weighted norm inequalities are proved for a rough homogeneous singular integral operator and its corresponding maximal truncated singular operator. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals.

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Boundedness of the Hilbert transform on Besov spaces

Carpathian Mathematical Publications ◽

10.15330/cmp.12.2.443-450 ◽

2020 ◽

Vol 12 (2) ◽

pp. 443-450

Author(s):

A. Maatoug ◽

S.E. Allaoui

Keyword(s):

Integral Operator ◽

Harmonic Analysis ◽

Singular Integral Operator ◽

Singular Integral ◽

Hilbert Transform ◽

Besov Spaces ◽

Singular Integrals ◽

Modern Theory ◽

The Hilbert Transform ◽

Positive Results

The Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on $L^p(\mathbb{R}^n)$ has been extensively studied by various authors in different contexts and the authors gave positive results for some or all $p,1<p<\infty$. Littlewood-Paley theory provides alternate methods for studying singular integrals. The Hilbert transform along curves, the classical example of a singular integral operator, led to the extensive modern theory of Calderón-Zygmund operators, mostly studied on the Lebesgue $L^p$ spaces. In this paper, we will use the Littlewood-Paley theory to prove that the boundedness of the Hilbert transform along curve $\Gamma$ on Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ can be obtained by its $L^p$-boundedness, where $ s\in \mathbb{R}, p,q \in ]1,+\infty[ $, and $\Gamma(t)$ is an appropriate curve in $\mathbb{R}^n$, also, it is known that the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ are embedded into $L^p(\mathbb{R}^n)$ spaces for $s >0$ (i.e. $B^{s}_{p,q}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), s>0)$. Thus, our result may be viewed as an extension of known results to the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ for general values of $s$ in $\mathbb{R}$.

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Weight Inequalities for Singular Integrals Defined on Spaces of Homogeneous and Nonhomogeneous Type

Georgian Mathematical Journal ◽

10.1515/gmj.2001.33 ◽

2001 ◽

Vol 8 (1) ◽

pp. 33-59

Author(s):

D. E. Edmunds ◽

V. Kokilashvili ◽

A. Meskhi

Keyword(s):

Integral Operator ◽

Singular Integral Operator ◽

Singular Integral ◽

Singular Integrals ◽

Sufficient Conditions ◽

Lorentz Spaces ◽

Weight Functions ◽

Weighted Lorentz Spaces

Abstract Optimal sufficient conditions are found in weighted Lorentz spaces for weight functions which provide the boundedness of the Calderón–Zygmund singular integral operator defined on spaces of homogeneous and nonhomogeneous type.

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