Related papers: Lipschitz and Triebel--Lizorkin spaces, commutator…

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_\theta$ (with $\theta$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their…

We characterize uniformly perfect, complete, doubling metric spaces which embed bi- Lipschitzly into Euclidean space. Our result applies in particular to spaces of Grushin type equipped with Carnot-Carath\'eodory distance. Hence we obtain…

In a previous paper two of the authors introduced and study Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha}(\gamma_{d})$ and Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha}(\gamma_{d})$. Now, in this paper we introduce the variable…

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spaces…

The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<\beta<\alpha$, any compact metric space $X$ of Hausdorff dimension $\alpha$…

We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\le k\le 2n+1$, let $\Delta_k$ denote the Hodge Laplacian restricted to $k$-forms. Our first…

This work provides an extension of parts of the classical finite dimensional sub-elliptic theory in the context of infinite dimensional compact connected metrizable groups. Given a well understood and well behaved bi-invariant Laplacian,…

It is well known that an (in general, non-commutative) set of non-Hermitian operators $\Lambda_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus…

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection $\mathcal C$ of Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in some…

The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $R^d$ associated with a root system, that contain some non-local refection terms, and the associated Hardy space is defined by means of the…

We examine some noncommutative spherically symmetric spaces in three space dimensions. A generalization of Snyder's noncommutative (Euclidean) space allows the inclusion of the generator of dilations into the defining algebra of the…

In the setting of $\R^d$ with an $n-$dimensional measure $\mu,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $\mu.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund…

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In…

We study Calder\'on-type commutators $[M_b,T_i\mathcal R_j]$ in the rational Dunkl setting with a finite reflection group $G$. If $b$ belongs to the orbit Lipschitz class $\operatorname{Lip}_d$, then for every $1<p<\infty$ we prove…

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean…

This paper provides equivalence characterizations of homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces, denoted by $\dot{F}^s_{p,q}(\mathbb{R}^n)$ and $\dot{B}^s_{p,q}(\mathbb{R}^n)$ respectively, in terms of maximal functions of the…

In this paper we prove the Sobolev embeddings for Herz-type Triebel-Lizorkin spaces, \begin{equation*} \dot{K}_{q}^{\alpha_{2},r}F_{\theta }^{s_{2}}\hookrightarrow \dot{K}%_{s}^{\alpha_{1},p}F_{\beta }^{s_{1}} \end{equation*} where the…

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base $\Omega\subset \mathbb{R}^d$ and a time-dependent separation $\Lambda$. Under certain mild regularity…

In a 2013 paper, Cheeger and Kleiner introduced a new type of dimension for metric spaces, the "Lipschitz dimension". We study the dimension-theoretic properties of Lipschitz dimension, including its behavior under Gromov-Hausdorff…

Motivated by the recent works [Huan Yu, Quansen Jiu, and Dongsheng Li, 2021] and [Yanping Chen and Zihua Guo, 2021], we study the following extension of Calder\'on-Zygmund type singular integrals $$ T_{\beta}f (x) = p.v. \int_{\mathbb{R}^n}…