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The aim of this work is to give sufficient conditions ensuring that the space PAP(𝕉, X, µ) of µ-pseudo almost periodic functions and the space PAA(𝕉, X, µ) of µ-pseudo almost automorphic functions are invariant by the convolution product f = k * f, k ∈ L 1(𝕉). These results establish sufficient assumptions on k and the measure µ. As a consequence, we investigate the existence and uniqueness of µ-pseudo almost periodic solutions and µ-pseudo almost automorphic solutions for some abstract integral equations, evolution equations and partial functional differential equations.

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^\Phi_\alpha(\mathbb B^n)$, which are generalizations of classical Bergman spaces. We obtain atomic decomposition for functions in the Bergman-Orlicz space $\mathcal A^\Phi_\alpha (\mathbb B^n)$ where $\Phi$ is either convex or concave growth function. We then prove weak factorization theorems involving the Bloch space and a Bergman-Orlicz space and also weak factorization theorems involving two Bergman-Orlicz spaces.

This work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). A multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism is proposed first. Indeed, there are human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self-infection by susceptible individuals. The global stability of the disease-free equilibrium is shown when a given threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{0} $$\end{document}T0 is less or equal to 1 and the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0} $$\end{document}R0 is calculated. A convergence index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{1} $$\end{document}T1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of infectious extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. Using Partial Rank Correlation Coefficient with a three levels fractional experimental design, the sensitivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0} $$\end{document}R0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{0} $$\end{document}T0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{1}$$\end{document}T1 to control parameters is evaluated. Following this study, the most significant parameter is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into account economic impacts of SARS-CoV-2, optimal fighting strategies are determined and discussed. The study is applied to real and available data from Cameroon with a model fitting. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy against SARS-CoV-2.

In this paper, we study the boundedness and the compactness of the little Hankel operators $$h_b$$ with operator-valued symbols b between different weighted vector-valued Bergman spaces on the open unit ball $$\mathbb {B}_n$$ in $$\mathbb {C}^n.$$ More precisely, given two complex Banach spaces X, Y, and $$0 < p,q \le 1,$$ we characterize those operator-valued symbols $$b: \mathbb {B}_{n}\rightarrow \mathcal {L}(\overline{X},Y)$$ for which the little Hankel operator $$h_{b}: A^p_{\alpha }(\mathbb {B}_{n},X) \longrightarrow A^q_{\alpha }(\mathbb {B}_{n},Y),$$ is a bounded operator. Also, given two reflexive complex Banach spaces X, Y and $$1< p \le q < \infty ,$$ we characterize those operator-valued symbols $$b: \mathbb {B}_{n}\rightarrow \mathcal {L}(\overline{X},Y)$$ for which the little Hankel operator $$h_{b}: A^p_{\alpha }(\mathbb {B}_{n},X) \longrightarrow A^q_{\alpha }(\mathbb {B}_{n},Y),$$ is a compact operator.

On the Lie ball $\omega$ of $\Bbb C^n$, $n\ge 3$, we prove that for all $p\in [1,\infty)$, $p\ne 2$, the Hardy space $H^p(\omega)$ is an uncomplemented subspace of the Lebesgue space $L^p (\partial_0\omega,d\sigma)$, where $\partial_0\omega$ denotes the Shilov boundary of $\omega$ and $d\sigma$ is a normalized invariant measure on $\partial_0\omega$.

We present a transference principle of Lebesgue mixed norm estimates for Bergman projectors from tube domains over homogeneous cones to homogeneous Siegel domains of type II associated to the same cones. This principle implies improvements of these estimates for homogeneous Siegel domains of type II associated with Lorentz cones, e.g. the Pyateckii-Shapiro Siegel domain of type II., Comment: 33 pages, 6 figures

The aim of this work is to establish several results on the existence and regularity of solutions for some nondensely nonautonomous partial functional differential equations with finite delay in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family under the conditions introduced by N. Tanaka.We show the local existence of the mild solutions which may blow up at the finite time. Secondly,we give sufficient conditions ensuring the existence of the strict solutions. Finally, we consider a reaction diffusion equation with delay to illustrate the obtained results.

In the setting of tube domains over symmetric cones, we determine a necessary and sufficient condition on a Borel measure \(\mu \) so that the Hardy space \(H^{p}, \ 1\le p < \infty ,\) continuously embeds in the weighted Lebesgue space \(L^q (T_\Omega ,d\mu )\) with a larger exponent. Finally we use this result to characterize multipliers from \(H^{2m}\) to Bergman spaces for every positive integer m.

The aim of this work is to study the existence of a periodic solution for some nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We use Massera’s approach (Duke Math 17:457–475, 1950), we prove that the existence of a bounded solution on \(\mathbb {R}^{+}\) implies the existence of an \(\omega \)-periodic solution. In nonlinear case, we use a fixed point for multivalued maps to show the existence of a periodic solution. Finally, we consider a reaction diffusion equation with delay to illustrate the main results of this work.

Starting from an adapted Whitney decomposition of tube domains in $\C^n$ over irreducible symmetric cones of $\R^n,$ we prove an atomic decomposition theorem in mixed norm weighted Bergman spaces on these domains. We also characterize the interpolation space via the complex method between two mixed norm weighted Bergman spaces., 27 pages

We show that the modulus of the Bergman kernel $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}B(z, \zeta )$ of a general homogeneous Siegel domain of type II is ‘almost constant’ uniformly with respect to $z$ when $\zeta $ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Korányi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used it to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces $A^p$ on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents $p$ via functional analysis using recent estimates.

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analogous to the dyadic partition $[2^j, 2^{ j + 1})$ of the positive real line, in this paper we develop an adapted Littlewood?Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p, q}_\nu$, where the role of the usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. We show that $B^{p, q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p, q}_\nu (T_\Omega)$, at least in a 'good range' of indices $1 \leq q 2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu \colon L^{p, q}_\nu \to A^{p, q}_\nu$, for which our result implies a positive answer when $q_{\nu, p}' < q < q_{\nu, p}$. This extends, to general cones, previous work of the authors on the light-cone. Finally, we focus on light-cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light-cones when $p$ is sufficiently large.

We prove that the complex interpolation space [ A ν p 0 , A ν p 1 ] θ , 0 θ A ν p 0 and A ν p 1 on the tube in C n , n ⩾3, over an irreducible symmetric cone of R n is the weighted Bergman space A ν p with 1/ p =(1− θ )/ p 0 + θ / p 1 . Here, ν > n / r −1 and 1⩽ p 0 p 1 ν /( n / r −1) where r denotes the rank of the cone. We then construct an analytic family of operators and an atomic decomposition of functions, which are related to this interpolation result. To cite this article: D. Bekolle et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

We prove that the functions of the Bergman spaces Ap on tube domains may be written as Laplace transforms of functions when 1 ≤ p ≤ 2. We give in this context a generalization of the Hausdorff–Young inequality with the exact constant, and deduce from the case p = 2 the expression of the Bergman kernel as a Laplace transform.

Molecular decompositions for functions in weighted Bergman spaces are extended to two homogeneous, non symmetric Siegel domains of type II, as well as interpolation by functions in weighted Bergman spaces. An ingredient in the proof is a generalization of a lemma due to Koranyi. We also specify the hypotheses in the symmetric case.

We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.

Let \(\Gamma\) be the future light cone in \({\Bbb R}^n\), and \(\Omega={\Bbb R}^n +i\Gamma\) be the associated tube domain. We prove that the weighted Bergman projection \(P_\nu\)

An affirmative answer is given to the following conjecture of R. Coifman and R. Rochberg: in any symmetric Siegel domain of type II, the dual of the Bergman space A 1 {A^1} coincides with the Bloch space of holomorphic functions and can be realized as the Bergman projection of L ∞ {L^\infty } .

We determine a defining kernel for the Bergman projection of L ∞ {L^\infty } in tubes over cones of real, symmetric, positive-definite matrices.

Soit Ω un domaine de Jordan à bord rectifiable du plan complexe C. Désignons par dλ la mesure de Lebesgue à l'intérieur de Ω, par dσ la mesure de Lebesgue sur le bord ∂Ω de Ω et par φ une représentation conforme de Ω sur le disque unité D du plan complexe.Par définition, la classe de Bergman Ap(Ω), 0 < p ≦ +∞, est le sous-espace de Lp(dλ) formé par les fonctions holomorphes dans Ω et le projecteur de Bergman PΩ de Ω est le projecteur orthogonal de L2(dλ) sur A2(Ω); quel que soit f dans L2(dλ), on a la formule:(1)