We give a proof of a so-called "local $Tb$" Theorem for singular integrals whose kernels satisfy the standard Calderón-Zygmund conditions. The present theorem, which extends an earlier result of M. Christ \cite{Ch}, was proved in \cite{AHMTT} for "perfect dyadic" Calderón-Zygmund operators. The proof in \cite{AHMTT} essentially carries over to the case considered here, with some technical adjustments.
We study the properties of reflectionless measures for an $s$-dimensional Calderón-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $μ$ for which $T(μ)$ is constant on the support of the measure. In this series of papers, we develop the basic theory of reflectionless measures, and describe the relationship between the description of reflectionless measures and certain well-known problems in harmonic analysis and geometric measure theory.
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 establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multil…
We study the properties of reflectionless measures for a Calderón-Zygmund operator T. Roughly speaking, these are measures $μ$ for which T(μ) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded i…
We give necessary and sufficient conditions for inhomogeneous Calderón-Zgymund operators to be bounded on the local hardy spaces $h^p(\mathbb{R}^n)$. We then give applications to local and truncated Riesz transforms, as well as pseudo-differential operators defined by amplitudes.
We are interested in a new kind of bi-dimensional bilinear paraproducts (appearing in [6]), which do not fit into the setting of bilinear Calderón-Zygmund operators. In this paper we propose a fiber-wise Calderón-Zygmund decomposition, which is specially adapted to this kind of bi-dimensional paraproduct.
We prove necessary and sufficient conditions for a Calderón-Zygmund operator to be compact at the endpoint from $L^{1}(\mathbb R^{d})$ into $L^{1,\infty}(\mathbb R^{d})$.
We continue our study of the reflectionless measures associated to an $s$-dimensional Calderón-Zygmund operator (CZO) acting in $\mathbb{R}^d$ with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $μ$ for which $T$ is a bounded operator in $L^2(μ)$. In this way, we shall reduce certain well-known problems at the interface of harmonic analysis and geometric measure theo…
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 prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators and generalized Radon transforms.
It is well-known that several classical results about Calderón-Zygmund singular integral operators can be extended to \(X\)-valued functions if and only if the Banach space \(X\) has the UMD property. The dependence of the norm of an \(X\)-valued Calderón-Zygmund operator on the UMD constant of the space \(X\) is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calderón-Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift 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…
In connection with proving the A_2 conjecture in 2010, T. Hytönen obtained a representation of general Caldéron-Zygmund operators in terms of simpler operators known as Haar shifts. In this note, we prove that the result is sharp in the sense that Haar shift representations of Hytönen's type are only available for Calderón-Zygmund operators.
We prove that a large class of operators, which arise as the projections of martingale transforms of stochastic integrals with respect to Brownian motion, as well as other closely related operators, are in fact Calderón--Zygmund operators. Consequently, such operators are not only bounded on $L^p$, $1<p<\infty$, but also satisfy weak-type inequalities. Unlike the boundedness on $L^p,$ which can be obtained directly from the Burkholder martingale transform inequalities, the weak-type estimates do not follow from the corresponding martingale results.
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.
In this paper, we introduce a class of singular integral operators which generalize Calderón-Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a general hyper curve. We show that such operators have similar properties as ordinary Calderón-Zygmund operators. In particular, we prove that they are of weak-type $(1, 1)$ and strong type $(p,p)$ for $1<p<\infty$.
In this note, we study the $A_p$-$A_\infty$ estimate for Calderón-Zygmund operators in terms of the weak $A_\infty$ characteristics in spaces of homogeneous type. The weak $A_\infty$ class was introduced recently by Anderson, Hytönen and Tapiola. Our estimate is new even in the Euclidean space.
Let T be an arbitrary L^2 bounded Calderon--Zygmund operator, and T_# its maximal truncated version. Then T_# satisfies the following bound for all 1<p<\infty and all weights w\in A_p: \|T_# \|_{L^p(w)} << [w]_{A_p}^{1/p} {[w]_{A_infty}^{1/p'}+[w^{1-p'}]_{A_infty}^{1/p}}.
Given a Calderón--Zygmund (C--Z for short) operator $T$, which satisfies Hörmander condition, we prove that: if $T$ maps all the characteristic atoms to $WL^{1}$, then $T$ is continuous from $L^{p}$ to $L^{p}(1<p<\infty)$. So the study of strong continuity on arbitrary function in $L^{p}$ has been changed into the study of weak continuity on characteristic functions.
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…
Let $μ$ be an $n$-dimensional finite positive measure on $\mathbb{R}^m$. We obtain a $T1$ condition sufficient for the boundedness of Calderón-Zygmund operators on $\textrm{RBMO}(μ)$, the regular BMO space of Tolsa.
Let $T$ be a Calderón-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,κ}(w)$ under appropriate conditions on the weight $w$, where the symbol $b$ belongs to weighted $BMO$ or Lipschitz space or weighted Lipschitz space.
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.
We prove that for a Calderon-Zygmund operator and weight w\in A_2, that it satisfies the linear in A2 bound due to Hytonen. Our proof will appeal to a distributional inequality used by several authors, adapted Haar functions, and standard stopping times.
We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to p…
We consider $\ell^r$ extensions of Calderon-Zygmund operators on weighted spaces $L^p(w)$ with $w$ an $A_p$ weight and $1 < p < \infty$. We give quantitative estimates of these operators' norm in terms of a given weight's $A_p$ characteristic. In particular, using a decomposition theorem of A. Lerner, we show that these extensions recover certain scalar estimates. Related results on general Banach spaces have been studied by Hytonen and Hannien.
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.
Here we show that Lerner's method of local mean oscillation gives a simple proof of the $A_2$ conjecture for spaces of homogeneous type: that is, the linear dependence on the $A_2$ norm for weighted $L^2$ Calderon-Zygmund operator estimates. In the Euclidean case, the result is due to Hytönen, and for geometrically doubling spaces, Nazarov, Rezinikov, and Volberg obtained the linear bound.
In this paper we obtain the weak type (1,1) boundedness of Calderon-Zygmund operators acting over operator-valued functions. Our main tools for its solution are a noncommutative form of Calderon-Zygmund decomposition in conjunction with a pseudo-localization principle for singular integrals, which is new even in the classical setting and of independent interest. Perhaps because of the hidden role of pseudo-localization and almost orthogonality, this problem has remained open for quite some time. We also consider Calderon-Zygmund operators associated to certain operator-valued kernels.
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.
We present new results on the two-weighted boundedness of singular integral operators and $L^p$ boundedness of the Orlicz maximal function. Namely, we extend a theorem of Pérez regarding the necessary and sufficient conditions for the boundedness of the Orlicz maximal function as well as give a new sufficient two-weighted boundedness assumption for Calderón-Zygmund singular integrals.
The aim of this paper is to get the boundedness of the commutators of multi-sublinear operators generated by local campanato functions and multilinear Calderón-Zygmund operators on the product generalized local Morrey spaces.
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…
In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones' $β$ numbers. We also use these new coefficients to prove that n-dimensional Calderon-Zygmund operators with odd kernel of type $C^2$ are bounded in $L^2(μ)$ if $μ$ is an n-dimensional uniformly rectifiable measure.
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…
In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\Mm$ introduced in \cite{LOPTT} and for multilinear Calderón-Zygmund operators. In particular we obtain a sharp mixed "$A_p-A_{\infty}$" bound for $\Mm$, some partial results related to a Buckley-type estimate for $\Mm$, and a sufficient condition for the boundedness of $\Mm$ between weighted $L^p$ spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calderón-Zygmund operators in terms of dyadic positive multilinear operators in the spirit…
We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander $S^{m}_{ρ, δ}$ classes. These results are new in the case $ρ< 1$, that is, outwith the scope of multilinear Calderón-Zygmund theory.
We investigate the weighted bounds for multilinear maximal functions and Calderón-Zygmund operators from $L^{p_1}(w_1)\times...\times L^{p_m}(w_m)$ to $L^{p}(v_{\vec{w}})$, where $1<p_1,...,p_m<\infty$ with $1/{p_1}+...+1/{p_m}=1/p$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. We prove the sharp bound for the multilinear maximal function for all such $p_1,..., p_m$ and prove the sharp bound for $m$-linear Calderón-Zymund operators when $p\geq 1$.
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 the setting of $\R^d$ with an $n-$dimensional measure $μ,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $μ.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $μ$ for which T(1) is the Lipschitz class $0.$
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.
We study several fundamental operators in harmonic analysis related to Jacobi expansions, including Riesz transforms, imaginary powers of the Jacobi operator, the Jacobi-Poisson semigroup maximal operator and Littlewood-Paley-Stein square functions. 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. Our proofs rely on an explicit formula for the Jacobi-Poisson kernel, which we derive from a product formula for Jacobi polynomials.
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.
The recent proof of the sharp weighted bound for Calderón-Zygmund operators has led to much investigation in sharp mixed bounds for operators and commutators, that is, a sharp weighted bound that is a product of at least two different $A_p$ weight constants. The reason why these are sought after is that the product will be strictly smaller than the original one-constant bound. We prove a variety of these bounds in spaces of homogeneous type, using the new techniques of Lerner, for both operators and commutators.
We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calderón-Zygmund operators as several functions can now depend on the same one-dimensional variable. The study of this class is motivated by examples related to the two-dimensional bilinear Hilbert transform and to bilinear ergodic averages. This paper is a sequel to the prior paper arXiv:1108.0917 by the first author.
We consider L^p two weight inequalities for maximal truncations of dyadic Calderon-Zygmund operators. In the case of one weight being doubling, a characterization is given, and for the general case, sufficient conditions are given, including standard and non-standard testing conditions. The arguments of this paper parallel the authors prior work arXiv:0805.4711, but in this paper are much easier.
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.
