We study two-weight weak-type estimates for the operator $S_v f = \mathcal{T}(fv)/v$, where $\mathcal{T}$ is the Hardy-Littlewood maximal operator or a Calder\'on-Zygmund operator (CZO) and $v$ is a weight. Concretely, under certain conditions on the weights involved, we prove that $S_v$ is bounded from $L^{1}(wv)$ to $L^{1,\infty} (uv)$. We also consider the corresponding inequalities when $\mathcal{T}$ is a higher-order commutator of a CZO. These types of results are inspired by the article of Sawyer in [21], (see also [17]).
In this paper, we establish a Fefferman--Stein inequality in terms of area function and non-tangential maximal function associated with the Schr\"odinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ denotes the sub-Laplacian on $\mathcal G$ and $V$ is a nonnegative locally...
We establish sharp weak-type bounds for commutators of Calder\'on--Zygmund operators and $\text{BMO}(\mathbb{R}^n)$ symbols with respect to $A_p$ weights. We prove that the bound \looseness=-1 $$ \|[b,T]\|_{L^p(w)\rightarrow L^{p,\infty}(w)}\lesssim [w]_{A_p}^{\max(2,p')} $$ for $p \in (1,\infty)$ and $w \in A_p$ is sh...
A. Mair, Oscar Andres Higuera Ramirez, Cody B. Stockdale· 0 citations
We prove that the centered Hardy--Littlewood maximal operator is not of weak type $(p,p)$ for any $1\le p<\infty$ in three solvable settings: the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ with the switch--walk--switch word metric, the Baumslag--Solitar groups $BS(1,n)$, $n\ge2$, with their digit word me...
In this paper, we establish optimal weighted norm inequalities for commutators of singular integral operators with rough kernels. While classical Calder\'{o}n-Zygmund theory relies heavily on pointwise gradient smoothness, we operate under the strictly weaker log-Dini regularity condition assumed merely on the $L^{1}\l...
The purpose of this paper is twofold. First, we investigate the Hardy-Littlewood type phenomena for Dirichlet solutions to the M\"obius invariant Laplace equation on the unit ball in $\mathbb{R}^n$. Our work extends and improves several key results due to Pavlov\'c [Rev. Mat. Iberoam. 23: 831-845, 2007] and Chen et al....
Jiaolong Chen, Shao-Lin Chen, H. Hamada et al.· 0 citations
The Calder\'on-Zygmund theory establishes the boundedness of singular integral operators on $L^p$ spaces for $1<p<\infty$, yet it encounters a failure at the endpoint $p = 1$. While radial counterexamples in $\mathbb{R}^n$ are well-documented, Pan-Shao-Wang-Wu \cite{psww2026} has showed that every nonconstant holomorph...
Hong-Rong Chen, Guo-Kuan Shao, Ju-Jie Wu et al.· 0 citations
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