Domains of type 1,1 operators: a case for Triebel–Lizorkin spaces

Pseudo-differential operators of type $1\text{,}1$ are proved continuous from the Triebel–Lizorkin space $F_{p\text{,}1}^d$ to $L_p$, $1?p<\infty$, when of order d, and this is, in general, the largest possible domain among the Besov and Triebel–Lizorkin spaces. Hörmander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators. To cite this article: J. Johnsen, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

The harmonic measure and a Reuter-type result for a process with Bessel components

Let $B_{p\text{,}q}$ be the $R^2$-valued process $(B_p\text{,}{B}_q)$ with independent Bessel components $B_p$ and $B_q$ with indices p and q strictly positive. In this paper we compute explicitly the law of the hitting time and place of a circle, centered at the origin, when $B_{p\text{,}q}$ starts from the center and deduce a Reuter-type independence result. We use mainly analytical tools from PDE theory. To cite this article: A. Ziadi, A. Bencherif-Madani, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for $N\ge 2$. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}{L}_q$ regularity class. Our approach is based on the maximal $L_p\text{-}{L}_q$ regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.     

A new method for lower bounds of L-functions

Let $L(s\text{,}\pi \text{,}r)$ be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for $L(s\text{,}\pi \text{,}r)$ when $\mathfrak{R}(s)=1$ using Eisenstein series. This method is applicable even when $L(s\text{,}\pi \text{,}r)$ is not known to be absolutely convergent for $\mathfrak{R}(s)>1$. To cite this article: S.S. Gelbart et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Le nombre des diviseurs unitaires d'un entier dans les progressions arithmétiques

Let the functions $d_{k\text{,}l}^{*}(n)$ and $d_{k\text{,}l}(n)$ be number of unitary divisors (see below) and number of divisors n in arithmetic progressions $\{l+mk\}$; k and l are integers relatively prime such that $1?l?k$ and let, for $n?2$

$F(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}(n))\ln (\phi (k)\ln n)}{\ln n}\text{,}{F}^{*}(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}^{*}(n))\ln (\phi (k)\ln n)}{\ln n}\text{and}$

$D^{*}(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}(n)/d_{k\text{,}l}^{*}(n))\ln (\phi (k)\ln n)}{\ln n}\text{,}$

where $\phi (k)$ is Euler's totient. The function $F(n\text{;}k\text{,}l)$ has been studied in [A. Derbal, A. Smati, C. A. Acad. Sci. Paris, Ser. I 339 (2004) 87–90]. In this Note we study the functions $F^{*}(n\text{;}k\text{,}l)$ and $D^{*}(n\text{;}k\text{,}l)$. We give explicitly their maximal orders and we compute effectively the maximum of $F^{*}(n\text{;}k\text{,}l)$ for $k=1\text{,}2\text{,}3$ and that of $D^{*}(n\text{;}k\text{,}l)$ for $k=1\text{,}3\text{,}5\text{,}7\text{,}8\text{,}9\text{,}10\text{,}11\text{,}13$. To cite this article: A. Derbal, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Wavelet optimal estimations for a density with some additive noises

Using wavelet methods, Fan and Koo study optimal estimations for a density with some additive noises over a Besov ball $B_{r,q}^s(L)(r,q⩾1)$ and over $L^2$ risk (Fan and Koo, 2002 [13]). The $L^{\infty }$ risk estimations are investigated by Lounici and Nickl (2011) [19]. This paper deals with optimal estimations over $L^p(1⩽p⩽\infty )$ risk for moderately ill-posed noises. A lower bound of $L^p$ risk is firstly provided, which generalizes Fan–Koo and Lounici–Nickl's theorems; then we define a linear and non-linear wavelet estimators, motivated by Fan–Koo and Pensky–Vidakovic's work. The linear one is rate optimal for $r⩾p$, and the non-linear estimator attains suboptimal (optimal up to a logarithmic factor). These results can be considered as an extension of some theorems of Donoho et al. (1996) [10]. In addition, our non-linear wavelet estimator is adaptive to the indices s, r, q and L.