Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing \(\mathsf {L}^{p}\)-bounds on the second fundamental form of an isometric immersion from \(\mathsf {L}^{p}\)-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.     

Calderón–Zygmund Operators on Weighted Hardy Spaces

We study the boundedness of Calderón–Zygmund operators on weighted Hardy spaces $H^p_w$ using Littlewood-Paley theory. It is shown that if a Calderón–Zygmund operator T satisfies T ^*1?=?0, then T is bounded on $H^p_w$ for $w\in A_{p(1+\frac\varepsilon n)}$ and $\frac n{n+\varepsilon}<p\le1$ , where ε is the regular exponent of the kernel of T.     

Calderón–Zygmund operators in the Bessel setting

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

An Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators

Journal of Fourier Analysis and Applications - The purpose of this article is to provide an alternative proof of the weak-type $$\left( 1,\ldots ,1;\frac{1}{m}\right) $$ estimate for m-multilinear...     

Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

Calderón–Zygmund Type Singular Operators in Weighted Generalized Morrey Spaces

We find conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces \(\mathcal {L}^{p,\varphi }(\mathbb {R}^n,w),\) defined by a function \(\varphi (x,r)\) and radial type weight \(w(|x-x_0|), x_0\in {\mathbb {R}}^{n}.\) These conditions are given in terms of inclusion into \(\mathcal {L}^{p,\varphi }(\mathbb {R}^n,w),\) of a certain integral constructions defined by \(\varphi \) and w. In the case of \(\varphi =\varphi (r)\) we also provide easy to check sufficient conditions for that in terms of indices of \(\varphi \) and w.     

Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems

?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      18. Hardy Space Estimates for Littlewood–Paley–Stein Square Functions and Calderón–Zygmund Operators      Jarod Hart  Guozhen Lu 《Journal of Fourier Analysis and Applications》2016,22(1):159-186       19. The boundedness of multilinear Calderón–Zygmund operators on Hardy spaces      JIZHENG HUANG  YU LIU 《Proceedings Mathematical Sciences》2013,123(3):383-392 In this paper, we study the boundedness of the multilinear Calderón–Zygmund operators on products of Hardy spaces.      20. Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators          下载免费PDF全文 Yan Lin  Guo Ming Zhang 《数学学报(英文版)》2016,32(11):1297-1311 In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.

The boundedness of multilinear Calderón–Zygmund operators on Hardy spaces

In this paper, we study the boundedness of the multilinear Calderón–Zygmund operators on products of Hardy spaces.     

Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators

In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.