We provide kernel estimates for pseudo-differential operators in the $S(m,g)$ Weyl-H\"ormander calculus classes. These new results lead to boundedness theorems like the weak (1,1) type for these classes and the boundedness of these operators from Hardy spaces to Lebesgue spaces $L^p$. We prove that under a suitable hypoelliticity condition on the weigh $m,$ one can get the weak (1,1) inequality for the Fefferman order established previously by the second author in analogy to the classical $L^p$-boundedness theorem due to Fefferman. Our approach makes an extension to $S(m,g)$ classes of previous estimates proved by \'Alvarez and Hounie, Nagase, Kumano-Go, etc, for the H\"ormander classes including new regularity properties in this framework from the Hardy space $H^p$ to $L^p$ and also on Lorentz spaces. We have identified that the {\it Uniform H\"ormander Condition} plays a fundamental role in some of our boundedness theorems.
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 2 citations
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
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
Let $n \ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematic...
We consider nonlocal divergence-form operators of the type $$PV\int_{\mathbb{R}^n}(u(x)-u(y))K(x, y) dy=0\quad \text{in} \; B_2,$$ where the kernel $K$ is symmetric ($K(x, y) = K(y, x)$). We first show that there exists a kernel satisfying the conditions of \cite[Open Question 2.1]{FRRO24} for which this problem admits...
We prove mixed $A_p$-$A_\infty$ estimates for sparse operators in two non-doubling settings. In the first setting, we consider sparse operators defined using stopping times in continuous time. We obtain both strong- and weak-type bounds with the same powers of the weight characteristics as in the classical setting. Som...
F. Gonçalves, Emiel Lorist· 0 citations
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