Paco Villarroya
We prove several off-diagonal and pointwise estimates for singular integral operators that extend compactly on $L^{p}(\mathbb R^{n})$.
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We improve bump conditions for the two-weight boundedness of Calderón-Zygmund operators introduced recently by R. Rahm and S. Spencer.
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 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.
We introduce a Calderon Zygmund decomposition such that the bad function has vanishing integral against a number of pure frequencies. Then we prove a variation norm variant of a maximal inequality for several frequencies due to Bourgain. To obtain the full range of Lp estimates we apply the multi frequency Calderon Zygmund decomposition.
In this work, we describe several results exhibited during a talk at the El Escorial 2012 conference. We aim to pursue the development of a multi-frequency Calderon-Zygmund analysis introduced in [9]. We set a definition of general multi-frequency Calderon-Zygmund operator. Unweighted estimates are obtained using the corresponding multi-frequency decomposition of [9]. Involving a new kind of maximal sharp function, weighted estimates are obtained.
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.
In this short note, we extend a local $Tb$ theorem that was proved in \cite{GHO} to a full multilinear local $Tb$ theorem.
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 $T$ 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 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 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.
Multilinear fractional Calderón-Zygmund operators with Dini type kernel
本文研究具有Dini型核的多线性分数次Calderón-Zygmund算子，证明其端点弱型估计，并在变指数Lebesgue空间上建立有界性。
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.
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…
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.
Sparse bounds for rough non-translation-invariant Calderón-Zygmund operators
本文针对带有粗糙非平移不变Calderón-Zygmund核的奇异积分算子建立稀疏界，通过适配非平移不变情形的Littlewood-Paley分析改进Seeger微局部分解，验证稀疏控制框架的局部化假设，进而得到Calderón型交换子等的定量加权估计。
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 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.
For a general Calderon-Zygmund operator $T$ on $R^N$, it is shown that $\|Tf\|_{L^2(w)}\leq C(T)\|w\|_{A_2}\|f\|_{L^2(w)}$ for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Perez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov-Treil-Volberg method of random dyadic systems with just one random system and completely without bad parts; (ii) a resulting representation of a general Calderon-Zygm…
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…
Boundedness of commutators for multilinear Calderón-Zygmund operators on Generalized Morrey Spaces
研究多线性Calderón-Zygmund算子与广义Campanato空间符号生成的交换子，给出其在具可变增长条件的广义Morrey空间上有界性的充要条件。
Calderón-Zygmund operators in the Bessel setting
本文研究了 Bessel 算子相关的调和分析基本算子，证明其为齐次型空间意义下的 Calderón-Zygmund 算子，从而由一般理论导出其映射性质。
Calderón-Zygmund operators and endpoint spaces for Hermite expansions
本文研究了与Hermite算子相关的Calderón-Zygmund型算子，建立了其在Hardy空间和Lipschitz空间等端点空间上有界的充要条件，并应用于分析Riesz变换和伪乘子的有界性。
Given a Calderón-Zygmund operator $T$, a classic result of Coifman-Rochberg-Weiss relates the norm of the commutator $[b, T]$ with the BMO norm of $b$. We focus on a weighted version of this result, obtained by Bloom and later generalized by Lacey and the authors, which relates $\| [b, T] : L^p(\mathbb{R}^n; μ) \to L^p(\mathbb{R}^n; λ) \|$ to the norm of $b$ in a certain weighted BMO space determined by $A_p$ weights $μ$ and $λ$. We extend this result to higher iterates of the commutator and recover a one-weight result of Chung-Pereyra-Perez in the process.
Revisiting mixed weak inequalities of Fefferman-Stein type for commutators of Calderón-Zygmund operators: an improvement
本文针对Calderón-Zygmund算子及其交换子，通过构造关于恰当测度的强Fefferman-Stein型不等式，建立了改进的Fefferman-Stein型混合弱不等式，推广了已有经典结果。
In this paper, we study the weighted estimates for multilinear Calderón-Zygmund operators %with multiple $A_{\vec{P}}$ weights from $L^{p_1}(w_1)\times...\times L^{p_m}(w_m)$ to $L^{p}(v_{\vec{w}})$, where $1<p, p_1,...,p_m<\infty$ with $1/{p_1}+...+1/{p_m}=1/p$ and $\vec{w}=(w_1,...,w_m)$ is a multiple $A_{\vec{P}}$ weight. We give weak and strong type weighted estimates of mixed $A_p$-$A_\infty$ type. Moreover, the strong type weighted estimate is sharp whenever $\max_i p_i \le p'/(mp-1)$.
Local $Tb$ theorems with $L^p$ type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. Until very recently, local $Tb$ theorems in the non-homogeneous case had only been proved assuming scale invariant ($L^{\infty}$ or BMO) testing conditions. The combination of non-scale-invariance and general measures is a delicate issue. In a previous paper we overcame this obstacle in the model case of square functions defined using general measures. In this paper we finally tackle the very demanding case of Calderón-Zygmund operators. That is, …
For an L ^2-bounded Calderon-Zygmund Operator T, and a weight w \in A_2, the norm of T on L ^2 (w) is dominated by A_2 characteristic of the weight. The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973, has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A_2 character of the weight can be exactly once in the proof. Accordingly, a large part of the proof uses two-weight technique…
We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices $α$ in this context. This generalizes a simpler method existing in the literature, but being valid for a restricted range of $α$. As an application, we prove that several fundamental operators in harmonic analysis of the Laguerre expansions, including maximal operators related to the heat and Poisson semigroups, Riesz transforms, Littlewood-Paley-Stein type square functions and multipliers of Laplace and Laplace-Stieltjes tra…
In recent years, it has been well understood that a Calderón-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.
Let $({\mathcal X}, d, μ)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition and the non-atomic condition that $μ(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(μ)$ is equivalent to that of $T$ on $L^p(μ)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(μ)$ to $L^{1,\,\infty}(μ).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(μ)$, then its maximal operator is bounded on $L^p(μ)$ for all $p\in (1, \infty…
Generalized Calderón-Zygmund operators on the Hardy space $H^1_ρ(\mathcal X)$
本文建立了一类新的log-Dini型广义Calderón-Zygmund算子在Yang和Zhou引入的Hardy空间$H^1_ρ(\mathcal X)$上有界性的充要条件，推广并统一了近期相关结果。
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