Algebras of Almost Periodic Functions with Bohr-Fourier Spectrum in a Semigroup: Hermite Property and its Applications

Holomorphic Sobolev spaces, Hermite ans special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

The images of Hermite and Laguerre Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterised. These are used to characterise the Schwartz class of rapidly decreasing functions. The image of the space of all tempered distributions is also considered and a Paley-Wiener theorem for the windowed Fourier transform is proved.

Uniformly convex and smooth Banach spaces and Lp-boundedness properties of Littlewood-Paley and area functions associated with semigroups

In this paper we obtain new characterizations of the uniformly convex and smooth Banach spaces. These characterizations are established by using Lp-boundedness properties of Littlewood-Paley functions and area integrals associated with heat semigroups and involving fractional derivatives. Our results apply for instance by considering the heat semigroups defined by Hermite and Laguerre operators that do not satisfy the Markovian property.

Fractional powers of the parabolic Hermite operator. Regularity properties

Let $\mathcal{L}= \partial_t- Δ_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $α>0$ define the Parabolic Hermite-Zygmund spaces $$ Λ^α_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\; {\rm and} \:\; \left\|\partial_y^k e^{-y\sqrt{\mathcal{L}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+α},\;\: {\rm with }\, k=[α]+1, y>0. \right\}, $$ with the obvious norm. It is shown that these spaces have a pointwise description of Hölder type. The fractional powers $\mathcal{L}^{\pm β}$ are well defined in these spaces and the following regularity propertie…

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$…

乘积空间上连续函数的延拓、Bohr紧化与殆周期函数

Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions

本文刻画了乘积空间上可连续延拓到其紧化的连续函数，并由此将任意拓扑结构的Bohr紧化实现为殆周期函数所成交换Banach代数的Gelfand空间，同时应用于殆周期函数空间之间等距映射的表示。

复函数代数的投射自由性与埃尔米特性

Projective freeness and Hermiteness of complex function algebras

本文研究了多种复函数代数的投射自由性与埃尔米特性，提出了极大理想空间上的新上同调充分条件，并基于 Borsuk 形状理论引入新代数类 $\mathscr{C}$，证明其投射张量积能保持这些性质。