Related papers: An operator-valued $T1$ theory for symmetric CZOs

In this series of two papers, we will prove a natural matrix weighted $T1$ theorem for matrix kernelled CZOs. In the current paper, we will prove matrix weighted norm inequalities for matrix symbolled paraproducts via a general matrix…

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 Calder\'on--Zygmund type…

This paper gives a systematic study of operator-valued local Hardy spaces. These spaces are localizations of the Hardy spaces defined by Tao Mei, and share many properties with Mei's Hardy spaces. We prove the ${\rm h}_1$-$\rm bmo$ duality,…

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calder\'on--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a…

We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators including Calder\'on-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order…

Let $T$ be the $\theta$-type Calder\'on-Zgymund operator with Dini condition. In this paper, we prove that for $b\in {\rm CMO}(\mathbb R^n)$, the commutator generated by $T$ with $b$ and the corresponding maximal commutator, are both…

In this paper, we focus on a class of fractional type integral operators that can be served as extensions of Riesz potential with kernels $$K(x,y)=\frac{\Omega_1(x-A_1 y)}{|x-A_1 y |^{\frac{n}{q_1}}} \cdots \frac{\Omega_m(x-A_m y)}{|x-A_m y…

We establish the equivalent characterisation of the weighted BMO space on the complex plane $\mathbb{C}$ via the two weight commutator of the Beurling--Ahlfors operator with a BMO function. Our method of proofs relies on the explicit kernel…

We establish the boundedness of the multilinear Calder\'on-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…

Let $T$ be a $m$-linear Calder\'{o}n-Zygmund operator of type $\omega$ with $\omega$ being nondecreasing and $\omega \in$ Dini(1) and $[\vec{b},\,T]$ be the commutator generated by $T$ with symbols $\vec{b}=(b_1,\,\ldots,\,b_m)$ belonging…

In this paper, the author studies the boundedness for a large class of sublinear operator $T_\alpha, \alpha\in[0,n)$ generated by Calder{\'o}n-Zygmund operators ($\alpha=0$) and generated by fractional integral operator ($\alpha>0$) on…

We establish weighted inequalities for $BMO$ commutators of sublinear operators for all $0<p<\infty$. For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse H\"{o}lder condition, we prove that $\bullet$…

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the…

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 $\theta$-type Calder\'on--Zygmund operators $T_\theta$ recently introduced in…

We give necessary and sufficient conditions for inhomogeneous Calder\'on-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…

In 1985, Bloom characterized the boundedness of the commutator $[b,H]$ as a map between a pair of weighted $L^{p}$ spaces, where both weights are in $A_p$. The characterization is in terms of a novel $BMO$ condition. We give a 'modern'…

Let $X$ be a metric space equipped with a metric $d$ and a nonnegative Borel measure $\mu$ satisfying the doubling property and let $\{\mathcal{A}_t\}_{t>0}$, be a generalized approximations to the identity, for example $\{\mathcal{A}_t\}$…

In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calder\'{o}n-Zygmund operators with a quantity of operators beyond…

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{\rho,\delta}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{\rho,\delta}^m) \subset…

In this article, we study the commutators of Hausdor? operator and establish their boundedness on weighted Herz space in the setting of Heisenberg group.