We study the Banach space and operator factorization structure of Janson-Sobolev and Janson-Sobolev-Hardy spaces. This new class of martingale spaces is determined by a $q$-adic filtration, a subspace $V\subset \mathbb{R}_0^{l\times q}$, and a rearrangement invariant function space $X$. Our main result shows that, for every bounded diagonal operator $D$, the operator $S = \sum_{t=1}^s \lambda_{\mathcal U}^{k_t}(D)Q_{\mathcal K_t}^{\mathcal B}$ determined by the linear functionals $\lambda_{\mathcal U}^{k_t}(D)$ and the canonical projections $Q_{\mathcal K_t}^{\mathcal B}$, almost projectionally factors through $D$ with constant $1^+$. As consequences, we obtain factorization results for diagonal operators both under a natural boundedness condition on the canonical projections and for all spaces equipped with the $L^1$-norm.
Let $\mathcal{H}$ be the separable, infinite-dimensional complex Hilbert space. In the first paper of this series, we introduced, by means of the Rosenblum operators, the shift representation operators $K_x=\sum_{n=0}^{\infty}T^nx\otimes e_n$, which relate an operator $T$ with $r(T)<1$ to the unilateral shift $S$ and r...
Jun-Sheng Fang, Bing-Zhe Hou, Chun-Lan Jiang et al.· 0 citations
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mat...
In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if...
R. Alvarado, Efstathios Konstantinos Chrontsios Garitsis· 0 citations
In this paper, we study Lipschitz spaces adapted to Dunkl--Schr\"odinger operators $\mathcal{L} = -\Delta_k + V$, where $\Delta_k$ is the Dunkl Laplacian and $V \ge 0$ belongs to the reverse H\"older class $\mathrm{RH}^q(dw)$ with $q>\max(1, \mathbf{N}/2)$. For $0<\beta<2$, we establish the full equivalence between the...
Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0<p\leq 1$. Specifically, \[ B^s_{p,q}(\Ome...
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
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