Variation operators for singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces

2022 ◽  
Author(s):  
Feng Liu ◽  
Peng Cui
Keyword(s):  
Sobolev Spaces ◽  
Singular Integrals ◽  
Morrey Spaces ◽  
Weighted Morrey Spaces ◽  
Variation Operators

scholarly journals Variation Inequalities for One-Sided Singular Integrals and Related Commutators

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 876
Author(s):  
Feng Liu ◽  
Seongtae Jhang ◽  
Sung-Kwun Oh ◽  
Zunwei Fu
Keyword(s):  
Singular Integrals ◽  
Morrey Spaces ◽  
Weighted Morrey Spaces ◽  
Variation Operators

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones.

scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

scholarly journals Variation inequalities related to Schrödinger operators on weighted Morrey spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 813-827
Author(s):  
Jing Zhang
Keyword(s):  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Variation Operators ◽  
Reverse Hölder

Abstract This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

scholarly journals Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Yali Pan ◽  
Changwen Li ◽  
Xinsong Wang
Keyword(s):  
Harmonic Analysis ◽  
Singular Integrals ◽  
Integral Operators ◽  
Oscillatory Integral ◽  
Morrey Spaces ◽  
Oscillatory Integrals ◽  
Weighted Morrey Spaces ◽  
Oscillatory Integral Operators ◽  
Oscillatory Singular Integrals

Oscillatory integral operators play a key role in harmonic analysis. In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on the weighted Morrey spacesLp,k(ω). Meanwhile, the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Sign in / Sign up

Export Citation Format

Share Document