We prove the dimension free estimates of the $L^p\to L^p$, 1< p ≤ ∞, norms of the Hardy-Littlewood maximal operator related to the optimal control balls on the Heisenberg group ℍⁿ.

We consider the maximal function $\left|\right|\left(S^{a}f\right)\left[x\right]{\left|\right|}_{L^{\infty }[-1,1]}$ where $\left(S^{a}f\right){\left(t\right)}^{\wedge }\left(\xi \right)=e^{{it\left|\xi \right|}^a}f̂\left(\xi \right)$ and 0 < a < 1. We prove the global estimate $\left|\right|S^{a}f{\left|\right|}_{L²(ℝ,L^{\infty }[-1,1])}{\le C\left|\right|f\left|\right|}_{H^s\left(ℝ\right)}$, s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

Dans cet article on s’intéresse à l’existence et l’unicité globale de solutions pour le système de Navier-Stokes à densité variable, lorsque la donnée initiale de la vitesse est dans l’espace de Besov homogène de régularité critique $B_{p,1}^{-1+\frac{N}{p}}\left({ℝ}^N\right)$. Notons que ce résultat fait suite aux résultats de H. Abidi qui a généralisé le travail de R. Danchin. Toutefois, dans les travaux antérieurs, l’existence de la solution est obtenue pour $1\&lt;p\&lt;2N$ et l’unicité est démontrée sous l’hypothèse plus restrictive $1\&lt;p\le N.$ Notre résultat...

One of the main results in modern harmonic analysis is the extrapolation theorem of J. L. Rubio de Francia for Ap weights. In this paper we discuss some recent extensions of this result. We present a new approach that, among other things, allows us to obtain estimates in rearrangement-invariant Banach function spaces as well as weighted modular inequalities. We also extend this extrapolation technique to the context of A∞ weights. We apply the obtained results to the dyadic square function. Fractional integrals, singular integral operators and their commutators with bounded mean oscillation functions are also considered. We present an extension of the classical results of Boyd and Lorentz-Shimogaki to a wider class of operators and also to weighted and vector-valued estimates. Finally, the same kind of ideas leads us to extrapolate within the context of an appropriate class of non A∞ weights and this can be used to prove a conjecture proposed by E. Sawyer.