Related papers: A unified method for maximal truncated Calder\'on-…

In this paper we recontextualize the theory of matrix weights within the setting of Banach lattices. We define an intrinsic notion of directional Banach function spaces, generalizing matrix weighted Lebesgue spaces. Moreover, we prove an…

We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$, where $x\in {\mathbb R}^n$ and $Q$ is a cube in ${\mathbb R}^n$. When applied to operators, this result recovers our recent works. On the other hand, our…

Let $p\in[1,\infty]$, $q\in(1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $\alpha\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calder\'on--Zygmund operator $T$ on…

We obtain weighted mixed inequalities for operators associated to a critical radius function. We consider Schr\"odinger Calder\'on-Zygmund operators of $(s,\delta)$ type, for $1<s\leq \infty$ and $0<\delta \leq 1$. We also give estimates of…

We demonstrate and develop dyadic-probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a…

We construct a slightly new noncommutative Calder\'on-Zygmund decomposition by further splitting the bad function. Using this tool, we prove the weak type (1,1) boundedness of noncommutative Calder\'on-Zygmund operators under a class of…

For the Hardy-Littlewood maximal and Calder\'on-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak…

We prove endpoint results for sparse domination of translation invariant multiscale operators. The results are formulated in terms of dilation invariant classes of Fourier multipliers based on natural localized $M^{p\to q}$ norms which…

Truncated Toeplitz operators are compressions of Toeplitz operators on model spaces; they have received much attention in the last years. This survey article presents several recent results, which relate boundedness, compactness, and…

A proof is given for the "only if" part of the result stated in the previous paper of the series that a suitably nondegenerate Calder\'on-Zygmund operator $T$ is bounded in a Banach lattice $X$ on $\mathbb R^n$ if and only if the…

The technique of sparse domination, i.e., dominating operators with sums of averages taken over sparsely distributed cubes, has seen rapid development recently within the realms of harmonic analysis. A useful extension of sparse domination…

Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{{\mathscr{D}},{\mathcal S}}$ by $${\mathcal A}_{{\mathscr{D}},{\mathcal…

The aim of this note is twofold. Firstly, we prove an abstract version of the Calder\'on transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does…

We give a new formulation of the $T1$ theorem for compactness of Calder\'on-Zygmund singular integral operators. In particular, we prove that a Calder\'on-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in…

We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all…

We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation…

In a previous work, "compact versions" of Rubio de Francia's weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces,…

Several properties of the Harnack domination of linear operators acting on Hilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators,…

A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach is proposed, providing one with criteria for their validity in the class of rearrangement-invariant…

We identify symmetric quasi-Banach range of the discrete Calder\'{o}n operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of…