Let $b_1,b_2\in \mathrm{BMO}\left({ℝ}^n\right)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfying the Sobolev regularity that ${sup}_{\kappa \in \mathbb{Z}}{\parallel {\sigma }_{\kappa }\parallel }_{W^{s_1,s_2}\left({ℝ}^{2n}\right)}<\infty$ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}\left({ℝ}^n\right)\times L^{p_2}\left({ℝ}^n\right)$$(p_1,...$

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 generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form $L^p\left(X\right)\subseteq \gamma \left(X\right)\subseteq L^q\left(X\right)$, in terms of the type p and cotype q of the Banach space X. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,...,r_n}$. It is proved that for any function $f\in W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r=n{({\sum }_{j=1}^{n}1/r_j)}^{-1}\le n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f(f\in C_0^{\infty },s=(s_1,...,s_n))$ the weighted norm $\parallel {\left(D^{s}f\right)}^{\wedge }{\parallel }_{L^1\left(w\right)}(w\left(\xi \right)={\left|\xi \right|}^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

We study one-dimensional oscillator integrals which arise as Fourier-Stieltjes transforms of smooth, compactly supported measures on smooth curves in Euclidean spaces and determine their decay at infinity, provided the curves satisfy certain geometric conditions.

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...

This article is concerned with the study of the discrete version of generalized ergodic Calderón-Zygmund singular operators. It is shown that such discrete ergodic singular operators for a class of superadditive processes, namely, bounded symmetric admissible processes relative to measure preserving transformations, are weak (1,1). From this maximal inequality, a.e. existence of the discrete ergodic singular transform is obtained for such superadditive processes. This generalizes the well-known...