In this paper, the author considers the weighted vector-valued estimate for the operator defined by $$T_Af(x)={\rm p.\,v.}\int_{\mathbb{R}^n}\frac{Ω(x-y)}{|x-y|^{n+1}}\big(A(x)-A(y)-\nabla A(y)\big)f(y){\rm d}y,$$ and the corresponding maximal operator $T_A^*$, where $Ω$ is homogeneous of degree zero, has vanishing moment of order one, $A$ is a function in $\mathbb{R}^n$ such that $\nabla A\in {\rm BMO}(\mathbb{R}^n)$. By a pointwise estimate for $\|\{T_Af_k(x)\}\|_{l^q}$ and the weighted $L^p$ estimates for the sparse operator $$\mathcal{A}_{\mathcal{S},\,L(\log L)^β}f(x)=\sum_{Q\in\mathcal{S}}\|f\|_{L(\log L)^β,\,Q}χ_{Q}(x) ,$$ the author establishes some weak and endpoint quantitative weighted vector-valued estimates for $T_A$ and $T_A^*$.
Two proofs of a weighted weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for multilinear Calderón-Zygmund operators are given. The ideas are motivated by different proofs of the classical weak-type $(1,1)$ estimate for Calderón-Zygmund operators. One proof uses the Calderón-Zygmund decomposition, and the other proof is motivated by ideas of Nazarov, Treil, and Volberg.
We prove necessary and sufficient conditions for a Calderón-Zygmund operator to be compact at the endpoint from $L^{1}(\mathbb R^{d})$ into $L^{1,\infty}(\mathbb R^{d})$.
In this article, we address pointwise sparse domination for multilinear Calderón-Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderón-Zygmund operators.
We study singular integral operators with kernels that are more singular than standard Calderón-Zygmund kernels, but less singular than bi-parameter product Calderón-Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T1 theorem and commutator estimates in a form reminiscent of the corresponding…
We present a new proof of the classical weak-type $(1,1)$ estimate for Calderón-Zygmund operators. This proof is inspired by ideas of Nazarov, Treil, and Volberg that address the non-doubling setting. An application to a weighted weak-type inequality is also given.
Let $p\in(1,\infty)$, $ρ\in (2, \infty)$ and $W$ be a matrix $A_p$ weight. In this article, we introduce a version of variation $\mathcal{V}_ρ({\mathcal T_n}_{\,,\,\ast})$ for matrix Calderón--Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the $L^p(W)$-boundedness of $\mathcal{V}_ρ({\mathcal T_n}_{\,,\,\ast})$ with norm \begin{align*} \|\mathcal{V}_ρ({\mathcal T_n}_{\,,\,\ast})\|_{L^p(W)\to L^p(W)}\leq C[W]_{A_p}^{1+{1\over p-1} -{1\over p}} \end{align*} by first proving a sparse domination of the variation of the scalar Calderón--Zygmund operator, a…
In this paper, we consider the boundedness properties of multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature. First, we prove strong type and weak type estimates for multilinear $θ$-type Calderón--Zygmund operators on products of weighted Morrey spaces with multiple weights. Then we discuss strong type estimates for both multilinear commutators and iterated commutators of $T_θ$ on products of these spaces with multiple weights. Furthermore, the weak end-point estimates for commutators of $T_θ$ and pointwise multiplication with functions in bounded mean …
It is well-known that dyadic martingale transforms are a good model for Calderón-Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that, if $W$ is an $A_2$ matrix weight, then the weighted $L^2$-norm of a Calderón-Zygmund operator with cancellation has the same dependence on the $A_2$ characteristic of $W$ as the weighted $L^2$-norm of the martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón-Zygmund operators on the $A_2$ characteristic of the weight is reduce…
We characterize the non-atomic measures $μ$ for which all Calderón-Zygmund operators with antisymmetric kernels of a fixed non-integer dimension $s$ are bounded in $L^2(μ)$ in terms of a positive quantity, the Wolff energy.
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 Calderón-Zygmund operators on $\mathbb{R}^n$ first proved by Grafakos and Torres. Subsequent proofs in the bilinear setting have been given by Maldonado and Naibo and also by Pérez and Torres. The proof given here is motivated by the proof of the weak-type $(1,1)$ estimate for Calderón-Zygmund operators in the nonhomogeneous setting by Nazarov, Treil, and Volberg.
In this note we will show a Calderón--Zygmund decomposition associated with a function $f\in L^1(\mathbb{T}^ω)$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.
In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{\infty}$ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón--Zygmund operators and th…
Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón--Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón--Zygmund o…
We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem.
In this note we give simple proofs of several results involving maximal truncated Caldeón-Zygmund operators in the general setting of rearrangement invariant quasi-Banach function spaces by sparse domination. Our techniques allow us to track the dependence of the constants in weighted norm inequalities; additionally, our results hold in $\mathbb{R}^n$ as well as in many spaces of homogeneous type.
In this paper, we first introduce several new classes of weighted amalgam spaces. Then we discuss both strong type and weak type estimates for certain multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature on products of these spaces with multiple weights. Furthermore, the strong type and weak end-point estimates for both multilinear commutators and iterated commutators of $T_θ$ and pointwise multiplication with BMO functions are established as well.
For a matrix A_2 weight W on R^p, we introduce a new notion of W-Calderón-Zygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit of Hytönen's Representation Theorem. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.
For a limited range of indices $p$, we obtain $L^p(\mathbb{R}^n)$ boundedness for singular integral operators whose kernels satisfy a condition weaker than the typical Hörmander smoothness estimate. These operators are assumed to be bounded (or weakly bounded) on $L^{s}(\mathbb{R}^n)$ for some index $s$. Our estimates are obtained via interpolation from the appropriate weak-type estimates. We provide two proofs of this result. One proof is based on the Calderón-Zygmund decomposition, while the other uses ideas of Nazarov, Treil, and Volberg.
Let $T$ be a multilinear Calderón-Zygmund operator of type $ω$ with $ω(t)$ being nondecreasing and satisfying a kind of Dini's type condition. Let $T_{Π\vec{b}}$ be the iterated commutators of $T$ with $BMO$ functions. The weighted strong and weak $L(\log{L})$-type endpoint estimates for $T_{Π\vec{b}}$ with multiple weights are established. Some boundedness properties on weighted variable exponent Lebesgue spaces are also obtained. As applications, multiple weighted estimates for iterated commutators of paraproducts and bilinear pseudo-differential operators with mild regularity are given.
In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases. Then we discuss the strong type and weak type estimates for a class of Calderón--Zygmund type operators $T_θ$ in these new Morrey type spaces. Furthermore, the strong type estimate and endpoint estimate of commutators $[b,T_θ]$ formed by $b$ and $T_θ$ are established. Also we study related problems about two-weight, weak type inequalities for $T_θ$ and $[b,T_θ]$ in the Morrey type spaces and give partial results.
In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we discuss the strong type and weak type estimates for a class of Calderón--Zygmund type operators $T_θ$ in these new weighted spaces. Furthermore, the strong type estimate and endpoint estimate of linear commutators $[b,T_θ]$ formed by $b$ and $T_θ$ are established. Also we study related problems about two-weight, weak type inequalities for $T_θ$ and $[b,T_θ]$ in the weighted amalgam spaces and give some results.
In this paper we study the relationship between two fundamental regularity properties of an $s$-dimensional Calderón-Zygmund operator (CZO) acting on a Borel measure $μ$ in $\mathbb{R}^d$, with $s\in (0,d)$. In the classical case when $s=d$ and $μ$ is equal to the Lebesgue measure, Calderón and Zygmund showed that if a CZO is bounded in $L^2$ then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower-dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical ha…
