scholarly journals Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Yanqi Yang ◽  
Shuangping Tao
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differentiation Operator ◽  
Fractional Differentiation ◽  
Herz Spaces ◽  
Variable Kernel ◽  
Harmonic Decomposition ◽  
Spherical Harmonic Decomposition ◽  
Fractional Differentiation Operator

Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ  (0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.

scholarly journals Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents

2018 ◽  
Vol 16 (1) ◽  
pp. 326-345 ◽  
Author(s):  
Yanqi Yang ◽  
Shuangping Tao
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Convolution Operator ◽  
Singular Integrals ◽  
Differentiation Operator ◽  
Variable Exponents ◽  
Fractional Differentiation ◽  
Variable Kernel ◽  
Fractional Differentiation Operator

AbstractLet T be the singular integral operator with variable kernel defined by $$\begin{array}{} \displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y \end{array} $$and Dγ(0 ≤ γ ≤ 1) be the fractional differentiation operator. Let T∗ and T♯ be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TDγ − DγT and (T∗ − T♯)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $ via the convolution operator Tm, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $(ℝn) is shown to hold for TDγ − DγT and (T∗ − T♯)Dγ. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1 ∘ T2.

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

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.

scholarly journals Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Amjad Hussain ◽  
Guilian Gao
Keyword(s):  
Singular Integral ◽  
Variable Exponent ◽  
Singular Integrals ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Herz Space ◽  
Herz Spaces ◽  
Multilinear Singular Integrals

The paper establishes some sufficient conditions for the boundedness of singular integral operators and their commutators from products of variable exponent Herz spaces to variable exponent Herz spaces.

Compactness of Commutators for Singular Integrals on Morrey Spaces

2012 ◽  
Vol 64 (2) ◽  
pp. 257-281 ◽  
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.

scholarly journals Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Vladyslav Litovchenko
Keyword(s):  
Moving Objects ◽  
Riesz Potential ◽  
Differentiation Operator ◽  
Local Interaction ◽  
General Nature ◽  
Pseudodifferential Equation ◽  
Fractional Differentiation ◽  
Gravitational Fields ◽  
Gravitational Influence ◽  
Fractional Differentiation Operator

Developing Holtzmark’s idea, the distribution of nonstationary fluctuations of local interaction of moving objects of the system with gravitational influence, which is characterized by the Riesz potential, is constructed. A pseudodifferential equation with the Riesz fractional differentiation operator is found, which corresponds to this process. The general nature of symmetric stable random Lévy processes is determined.

scholarly journals On a measure of non-compactness for singular integrals

2003 ◽  
Vol 1 (1) ◽  
pp. 35-43 ◽  
Author(s):  
Alexander Meskhi
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Lebesgue Space ◽  
Singular Integrals ◽  
Weighted Lebesgue Space ◽  
Smooth Curves

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.

scholarly journals Weighted norm inequalities for a class of rough singular integrals

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.

scholarly journals Boundedness of Singular Integral Operators with Variable Kernels on Weighted Weak Hardy Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Hua Wang
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Variable Kernel ◽  
Decomposition Theory

Let TΩ be the singular integral operator with variable kernel Ω(x,z). In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of TΩ on these spaces, under some Dini type conditions imposed on the variable kernel Ω(x,z).

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