Zygmund Operators Research Articles

Backward Uniqueness for 3D Navier–Stokes Equations With Non-Trivial Final Data and Applications

Abstract Presented is a backward uniqueness result of bounded mild solutions of 3D Navier–Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0, T]$ if it is so at time $T$. The proof is based on a new weighted estimate that enables to treat terms involving Calderon–Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.

Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces III: characterizations of molecules and wavelets, trace theorems, and boundedness of pseudo-differential operators and Calderón–Zygmund operators

Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces III: characterizations of molecules and wavelets, trace theorems, and boundedness of pseudo-differential operators and Calderón–Zygmund operators

Norm Inequalities for Calderón–Zygmund Operators in Some Generalized Hardy–Morrey Spaces

Norm Inequalities for Calderón–Zygmund Operators in Some Generalized Hardy–Morrey Spaces

Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces

Abstract In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.

Quantitative weighted estimates for generalized commutators of multilinear Calderón–Zygmund operators with the kernels of Dini type

Quantitative weighted estimates for generalized commutators of multilinear Calderón–Zygmund operators with the kernels of Dini type

Correction to: Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Correction to: Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

On the Predual of a Morrey–Lorentz Space and Its Applications to the Linear Calderón–Zygmund Operators

On the Predual of a Morrey–Lorentz Space and Its Applications to the Linear Calderón–Zygmund Operators

Commutators on Spaces of Homogeneous Type in Generalized Block Spaces

In this paper, we are interested in studying a generalized block space (denoted as Bφp,r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}^{p,r}_\\varphi $$\\end{document}) on a space of homogeneous type. We show that this space is the predual of certain generalized Morrey–Lorentz space. By duality, we obtain the Bφp,r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}^{p,r}_\\varphi $$\\end{document}-bound of operators of Calderón–Zygmund type. In addition, we prove a weak Hardy factorization in terms of commutators of integral operator of Calderón–Zygmund type in block spaces. Thanks to the Hardy factorization result, we obtain a characterization of functions in BMO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{BMO}$$\\end{document} via the boundedness of commutators of homogeneous linear Calderón–Zygmund operators in the generalized block space (resp. the generalized Morrey–Lorentz space). Finally, we study a compactness characterization of commutators of Calderón–Zygmund type in generalized Morrey–Lorentz spaces.

Weighted bilinear multiplier theorems in Dunkl setting via singular integrals

Abstract The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.

Two‐weight extrapolation on function spaces and applications

AbstractThis paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including , , and extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman–Fefferman inequalities that can be used to show some known sharp inequalities, Muckenhoupt–Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calderón–Zygmund operators. Finally, we obtain two‐weight inequalities for Littlewood–Paley operators and Fourier integral operators on weighted Banach function spaces.

Commutators of bilinear $$\theta $$-type Calderón–Zygmund operators on two weighted Herz spaces with variable exponents

Commutators of bilinear $$\theta $$-type Calderón–Zygmund operators on two weighted Herz spaces with variable exponents

Weak type estimates of genuine Calderón–Zygmund operators on the local Morrey spaces associated with ball quasi-Banach function spaces

Abstract Weak type estimates for genuine Calderón–Zygmund operators are established on the local Morrey spaces associated with ball quasi-Banach function spaces by two different methods. Above all, we obtain weak type estimates for the operator on the local weak Morrey spaces with variable exponents.

Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner–Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón–Zygmund operators, the previously known quantitative Bloom bounds are not sharp for the second and higher order commutators.

Bilinear Θ‐type Calderón–Zygmund operators and their commutators on product generalized fractional mixed Morrey spaces

AbstractThe aim of this paper is to investigate the boundedness of the bilinear θ‐type Calderón–Zygmund operator and its commutator on the product of generalized fractional mixed Morrey spaces. Under assumption that the positive and increasing functions defined on [0, ∞) satisfy doubling conditions, we prove that the bilinear θ‐type Calderón–Zygmund operator is bounded from the product of generalized fractional mixed Morrey spaces into spaces , where , , , for , and for . Furthermore, the boundedness of the commutator formed by and on spaces is also obtained.

Limited Range Extrapolation with Quantitative Bounds and Applications

In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of the A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} conjecture solved by Hytönen. Advances have greatly improved conceptual understanding of classical objects such as Calderón–Zygmund operators. However, plenty of operators do not fit into the class of Calderón–Zygmund operators and fail to be bounded on all Lp(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(w)$$\\end{document} spaces for p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (1, \\infty )$$\\end{document} and w∈Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w \\in A_p$$\\end{document}. In this paper we develop Rubio de Francia extrapolation with quantitative bounds to investigate quantitative weighted inequalities for operators beyond the (multilinear) Calderón–Zygmund theory. We mainly establish a quantitative multilinear limited range extrapolation in terms of exponents pi∈(pi-,pi+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_i \\in (\\mathfrak {p}_i^-, \\mathfrak {p}_i^+)$$\\end{document} and weights wipi∈Api/pi-∩RH(pi+/pi)′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w_i^{p_i} \\in A_{p_i/\\mathfrak {p}_i^-} \\cap RH_{(\\mathfrak {p}_i^+/p_i)'}$$\\end{document}, i=1,…,m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1, \\ldots , m$$\\end{document}, which refines a result of Cruz-Uribe and Martell. We also present an extrapolation from multilinear operators to the corresponding commutators. Additionally, our result is quantitative and allows us to extend special quantitative estimates in the Banach space setting to the quasi-Banach space setting. Our proof is based on an off-diagonal extrapolation result with quantitative bounds. Finally, we present various applications to illustrate the utility of extrapolation by concentrating on quantitative weighted estimates for some typical multilinear operators such as bilinear Bochner–Riesz means, bilinear rough singular integrals, and multilinear Fourier multipliers. In the linear case, based on the Littlewood–Paley theory, we include weighted jump and variational inequalities for rough singular integrals.

The Schatten Classes of Calderón–Zygmund Operators

We prove sufficient conditions for a Calderón–Zygmund operator to belong to the Schatten classes. As in the classical T1 theory, the conditions are given in terms of the smoothness of the operator kernel, and the action of both the operator and its adjoint on the function 1. To show membership to the Schatten class when p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>2$$\\end{document} we develop new bump estimates for composed Calderón–Zygmund operators, and a new extension of Carleson’s Embedding Theorem.

Weak‐type Fefferman–Stein inequality and commutators on weak Orlicz–Morrey spaces

AbstractWe consider the Fefferman–Stein inequality for weak Orlicz–Morrey spaces and the commutators and on weak Orlicz–Morrey spaces, where T is a Calderón–Zygmund operator, is a generalized fractional integral operator and b is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness from of and from a weak Orlicz Morrey space to another weak Orlicz–Morrey space. We use the Fefferman–Stein inequality to prove the boundedness of the commutators. Since weak Orlicz–Morrey spaces contain the weak Lebesgue, weak Orlicz and weak Morrey spaces as special cases, our results contain the bounedness on these function spaces which are also new results.

Endpoint Entropy Fefferman–Stein Bounds for Commutators

In this paper endpoint entropy Fefferman–Stein bounds for Calderón–Zygmund operators introduced by Rahm (J Math Anal Appl 504(1):Paper No. 125372, 2021) are extended to iterated Coifman–Rochberg–Weiss commutators.

Necessary and Sufficient Conditions for Commutator of the Calderón–Zygmund Operator on Mixed-Norm Herz-Slice Spaces

We obtain the separability of mixed-norm Herz-slice spaces, establish a weak convergence on mixed-norm Herz-slice spaces, and get the boundedness of the Calderón–Zygmund operator T on mixed-norm Herz-slice spaces. Moreover, we get the necessary and sufficient conditions for the boundedness of the commutator [b,T] on mixed-norm Herz-slice spaces, where b is a locally integrable function.