Related papers: Extrapolation for multilinear Muckenhoupt classes …

We develop a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. More specifically, if a $(D_{\theta}, \delta_1, \delta_{2, 3})$-Calder\'{o}n-Zygmund operator $T$ associated with Zygmund dilations…

Let $\delta\in(0,1]$ and $T$ be a $\delta$-Calder\'on-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+\delta/n}({\mathbb R}^n)$ satisfying $\int_{{\mathbb R}^n}\frac {w(x)}{1+|x|^n}\,dx<\infty$. When $b\in{\rm BMO}(\mathbb…

The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces $H^p(\mathbb{R}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts…

We prove that for operators satistying weighted inequalities with $A_p$ weights the boundedness on a certain class of Morrey spaces holds with weights of the form $|x|^\alpha w(x)$ for $w\in A_p$. In the case of power weights the shift with…

Commutators of bilinear pseudodifferential operators with symbols in the H\"ormander class BS_{1, 0}^1 and multiplication by Lipschitz functions are shown to be bilinear Calder\'on-Zygmund operators. A connection with a notion of…

It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^{p_1}(R) \times L^{p_2}(R) into L^{p_3}(R) whenever (p_1,p_2,p_3) is a Holder tuple with p_1,p_2 > 1 and p_3>2/3. We study the behavior of the quartile operator,…

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that…

We identify the optimal range of the Calder\`{o}n operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the optimal range of the triangular…

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 Calder\'on-Zygmund operators on $\mathbb{R}^n$ first proved by Grafakos and Torres.…

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous…

In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood…

We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we…

In this work, we extend Howard's construction of compatible families of Heegner points to the setting of towers of Gross curves and Shimura curves over totally real fields. Following the strategy of Longo and Vigni, our approach…

We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However,…

Given a Calder\'on-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…

Let $T$ be a bilinear Calder\'on-Zygmund singular integral operator and $T^*$ be its corresponding truncated maximal operator. For any $b\in\text{BMO}(\mathbb {R}^n)$ and $\vec{b}=(b_1,\ b_2)\in\text{BMO}(\mathbb {R}^n)\times\text…

This paper is a successor of \cite{laceyt}. In that paper we considered bilinear operators of the form H_alpha(f_1,f_2)(x) = p.v. \int f_1(x-t) f_2(x + alpha t)/t dt, which are originally defined for f_1, f_2 in the Schwartz class S(R). The…

This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the H\"{o}rmander class $BS_{\varrho,\delta}^m(\mathbb{R}^n)$ in the previously unexplored regime $0 \leq \varrho < \delta < 1$. We establish…

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the…

We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued…