Asymptotic boundedness and moment asymptotic expansion in ultradistribution spaces

An excursion to multiplications and convolutions on modulation spaces

We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.

On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $\ell^2(\mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(α)$ is non-negative if and only if $α$ is the moment sequence of a measure $μ$ on $\mathbb{R}^\infty$, assuming that $α$ does not grow too fast. We then characterize the non-negative bounded H…

Boundedness of Fractional Integrals on Special John--Nirenberg--Campanato and Hardy-Type Spaces via Congruent Cubes

Let $p\in[1,\infty]$, $q\in[1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors first find a reasonable version $\widetilde{I}_β$ of the (generalized) fractional integral $I_β$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_β$. Then the authors prove that $\widetilde{I}_β$ is bounded from $JN_{(p,q,s)_α}^…

分数阶Volterra型算子：从具有双侧加倍权的Bergman空间到Hardy空间

Fractional Volterra-type operators from Bergman spaces with two-sided doubling weights to Hardy spaces

本文给出了分数阶Volterra型算子从加倍权Bergman空间到Hardy空间有界性与紧性的充要符号条件，通过Littlewood-Paley理论与约化方法，最终条件仅依赖参数β。