Functional Analysis

New submissions

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New submissions for Mon, 13 May 24

[1]  arXiv:2405.06154 [pdf, ps, other]

Title: A lower bound for the Balan--Jiang matrix problem

Authors: Afonso S. Bandeira, Dustin G. Mixon, Stefan Steinerberger

Subjects: Functional Analysis (math.FA)

We prove the existence of a positive semidefinite matrix $A \in \mathbb{R}^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $\ell^1-$norm, more precisely $$ \sum_{k} x_k x_k^*=A \quad \implies \quad \sum_k \|x_k\|^2_{1} \geq c \sqrt{n} \|A\|_{1},$$ where $c$ is independent of $n$. This provides a lower bound for the Balan--Jiang matrix problem. The construction is probabilistic.

[2]  arXiv:2405.06210 [pdf, ps, other]

Title: Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz-Morrey spaces

Authors: Naoya Hatano

Subjects: Functional Analysis (math.FA)

It is known that the necessary and sufficient conditions of the boundedness of commutators on Morrey spaces are given by Di Fazio, Ragusa and Shirai. Moreover, according to the result of Cruz-Uribe and Fiorenza in 2003, it is given that the weak-type boundedness of the commutators of the fractional integral operators on the Orlicz spaces as the endpoint estimates. In this paper, we gave the extention to the weak-type boundedness on the Orlicz-Morrey spaces.

[3]  arXiv:2405.06453 [pdf, ps, other]

Title: Searching for linear structures in the failure of the Stone-Weierstrass theorem

Authors: Marc Caballer, Sheldon Dantas, Daniel L. Rodríguez-Vidanes

Subjects: Functional Analysis (math.FA)

We analyze the existence of vector spaces of large dimension inside the set $\mathcal{C}(L, \K) \setminus \overline{\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal C(L, \K)$ that vanishes nowhere on $L$ and does not necessarily separate the points of $L$. The results depend strongly on an equivalence relation that is defined on the algebra $\mathcal{A}$, denoted by $\sim_{\mathcal{A}}$, and a cardinal number that depends on $\sim_{\mathcal{A}}$ which we call the order of $\sim_{\mathcal{A}}$. We then introduce two different cases, when the order of $\sim_{\mathcal{A}}$ is finite or infinite. In the finite case, we show that $\mathcal{C}(L, \K) \setminus \overline{\mathcal{A}}$ is $n$-lineable but not $(n+1)$-lineable with $n$ being the order of $\sim_{\mathcal{A}}$. On the other hand, when the order of $\sim_{\mathcal{A}}$ is infinite, we obtain general results assuming, for instance, that the codimension of the closure of $\mathcal{A}$ is infinite or when $L$ is sequentially compact. To be more precise, we introduce the notion of the Stone-Weiestrass character of $L$ which is closely related to the topological weight of $L$ and allows us to describe the lineability of $\mathcal{C}(L, \K) \setminus \overline{\mathcal{A}}$ in terms of the Stone-Weierstrass character of subsets of $\sim_{\mathcal A}$. We also prove, in the classical case, that $(\mathcal{C}(\partial{D}, \C) \setminus \overline{\mbox{Pol}(\partial{D})}) \cup \{0\}$ (where $\mbox{Pol}(\partial{D})$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk) contains an isometric copy of $\text{Hol}(\partial{D})$ and is strongly $\mathfrak c$-algebrable, extending previous results from the literature.

Cross-lists for Mon, 13 May 24

[4]  arXiv:2405.04989 (cross-list from math.CV) [pdf, ps, other]

Title: Paley-Wiener Type Theorems associated to Dirac Operators of Riesz-Feller type

Authors: Swanhild Bernstein, Nelson Faustino

Comments: 36 pages

Subjects: Complex Variables (math.CV); Functional Analysis (math.FA)

This paper systematically investigates Paley-Wiener-type theorems in the context of hypercomplex variables. To this end, we introduce and study the so-called generalized Bernstein spaces endowed by the fractional Dirac operator $D_{\alpha}^{\theta}$ - a space-fractional operator of order $\alpha$ and skewness $\theta$, encompassing the Dirac operator $D$. We will show that such family of function spaces seamlessly characterizes the interplay between Clifford-valued $L^p-$functions satisfying the support condition $\mathrm{supp}\ \widehat{f}\subseteq B(0,R)$, and the solutions of the Cauchy problems endowed by the space-time operator $\partial_{x_0}+D_\theta^\alpha$ that are of exponential type $R^\alpha$. Such construction allows us to generalize, in a meaningful way, the results obtained by Kou and Qian (2002) and Franklin, Hogan and Larkin (2017). Noteworthy, the exploitation of the well-known Kolmogorov-Stein inequalities to hypercomplex variables permits us to make the computation of the maximal radius $R$ for which $\mathrm{supp}\ \widehat{f}$ is compactly supported in $B(0,R)$ rather explicit.

[5]  arXiv:2405.06200 (cross-list from math-ph) [pdf, ps, other]

Title: Restricted isometric compression of sparse datasets into low-dimensional varieties

Authors: Vasile Pop, Iuliana Teodorescu, Razvan Teodorescu

Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA); Optimization and Control (math.OC); Representation Theory (math.RT); Statistics Theory (math.ST)

This article extends the known restricted isometric projection of sparse datasets in Euclidean spaces $\mathbb{R}^N$ down into low-dimensional subspaces $\mathbb{R}^k, k \ll N,$ to the case of low-dimensional varieties $\mathcal{M} \subset \mathbb{R}^N,$ of codimension $N - k = \omega(N)$. Applications to structured/hierarchical datasets are considered.

[6]  arXiv:2405.06316 (cross-list from math.NA) [pdf, other]

Title: Refined localization spaces, Kondratiev spaces with fractional smoothness and extension operators

Authors: Markus Hansen, Cornelia Schneider

Subjects: Numerical Analysis (math.NA); Functional Analysis (math.FA)

In this paper, we introduce Kondratiev spaces of fractional smoothness based on their close relation to refined localization spaces. Moreover, we investigate relations to other approaches leading to extensions of the scale of Kondratiev spaces with integer order of smoothness, based on complex interpolation, and give further results for complex interpolation of those function spaces. As it turns out to be one of the main tools in studying these spaces on domains of polyhedral type, certain aspects of the analysis of Stein's extension operator are revisited. Finally, as an application, we study Sobolev-type embeddings.

[7]  arXiv:2405.06385 (cross-list from math.CV) [pdf, ps, other]

Title: Higher complex Sobolev spaces on complex manifolds

Authors: Thai Duong Do, Duc-Bao Nguyen

Subjects: Complex Variables (math.CV); Functional Analysis (math.FA)

We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex Monge-Amp\`{e}re equation.

Replacements for Mon, 13 May 24

[8]  arXiv:2208.14483 (replaced) [pdf, ps, other]

Title: Multiple products of meromorphic functions

Authors: A.Zuevsky

Subjects: Functional Analysis (math.FA)

[9]  arXiv:2308.13238 (replaced) [pdf, ps, other]

Title: Twisted shift preserving operators on $L^{2}(\mathbb{R}^{2n})$

Authors: Rabeetha Velsamy, Radha Ramakrishnan

Subjects: Functional Analysis (math.FA)

[10]  arXiv:2405.01133 (replaced) [src]

Title: A missing theorem on dual spaces

Authors: David P. Blecher

Comments: Error in early lemma (Proposition 2.3) endangers Theorem 4.1. Result is valid assuming the conclusions of Proposition 2.3 for the map q in Theorem 4.1

Subjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA); Operator Algebras (math.OA)

[11]  arXiv:2306.12887 (replaced) [pdf, ps, other]

Title: Consistent maps and their associated dual representation theorems

Authors: Charles L. Samuels

Subjects: Number Theory (math.NT); Functional Analysis (math.FA)

[12]  arXiv:2307.10889 (replaced) [pdf, ps, other]

Title: A nonlinear Strassen law for singular SPDEs

Authors: Shalin Parekh

Comments: 3rd version: fixed various small typos appearing in the published version

Journal-ref: Electron. J. Probab. 29 1 - 42, 2024.

Subjects: Probability (math.PR); Functional Analysis (math.FA)

• New submissions
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