Related papers: Limiting weak-type behavior for rough bilinear ope…

This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak type $(1,1)$ boundedness for noncommutative maximal operators with rough kernels. The proof of weak type (1,1)…

We establish the full quasi-Banach range of $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \rightarrow L^p(\mathbb R)$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $\Omega$ to the…

In this work, we establish $L^{p_1}\times \cdots\times L^{p_m}\to L^p$ bounds for maximal multi-(sub)linear singular integrals associated with homogeneous kernels $\frac{\Omega(\vec{\boldsymbol{y}}')}{|\vec{\boldsymbol{y}}|^{mn}}$ where…

Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$…

We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_\Omega$ associated with an integrable function…

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator $(I+L)^{-s…

Let $m\in \mathbb{N}$ and $0<\alpha<mn$.In this paper, we will use the idea of Hedberg to reprove that the multilinear operators $\mathcal{T}_{\Omega,\alpha;m}$ and $\mathcal{M}_{\Omega,\alpha;m}$ are bounded from $L^{p_1}(\mathbb…

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for…

In this paper, we study the boundedness properties of the (dyadic) maximal bilinear operator associated with rough homogeneous kernels on $\mathbb{R}$. We establish sharp $L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \to…

Let $M_{\Omega,\alpha}$ and $T_{\Omega,\alpha}$ be the fractional maximal and integral operators with rough kernels, where $0<\alpha<n$. In this paper, we shall study the continuity properties of $M_{\Omega,\alpha}$ and $T_{\Omega,\alpha}$…

In this paper, we investigate the boundedness of bilinear Calder\'on-Zygmund operators $T$ from ${L^{p_1}\left(w_1\right)} \times {L^{p_2}\left(w_2\right)}$ to ${L^{p,\infty}\left(v_{\vec{w}}\right)}$ with the stopping time method, where $1…

Let $T_\Omega$ be the singular integral operator with variable kernel $\Omega(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_\Omega$ on these…

The operators $\Lambda_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $\Lambda_0$ is equal to…

We establish analogs of sharp weighted weak-type bounds for $m$-sublinear operators satisfying sparse form domination, including multilinear Calder\'on-Zygmund singular integrals. Our results, which hold for general $\vec{p} \in…

Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\"older…

In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as…

The aim of this paper is to show that the discrete maximal function $$\mathcal{M}_{h}f(x)=\sup_{N\in\mathbb{N}}\frac{1}{|\mathbf{N}_{h}\cap[1, N]|}\Big|\sum_{n\in \mathbf{N}_{h}\cap[1, N]}f(x-n)\Big|,\ \ \mbox{for $x\in\mathbb{Z}$},$$ is of…

Let $r>\frac{4}{3}$ and let $\Omega \in L^{r}(\mathbb{S}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_{\Omega}(f,g)(x)= \textrm{p.v.}…

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(\Omega)/(M(\Omega)|\Omega|)$ and $M(\Omega)\lambda_1(\Omega) $, where $\Omega$ is a…

We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^\alpha$. This improves the previously best known condition…