In this paper, we characterize boundedness and compactness of weighted composition operators on the Dirichlet space and obtain the estimates for the essential norm.

In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{\varphi }$, when $\varphi$ is a linear-fractional self-map of $𝔻$. In this paper first, we investigate the essential normality problem for the operator $T_w{C}_{\varphi }$ on the Hardy space $H^2$, where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F\left(\varphi \right)$, $\varphi \in 𝒮\left(2\right)$, and $T_w$ is the Toeplitz operator with symbol $w$. Then we use these results and characterize the essentially normal...

We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^p$ and $H^q$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

We estimate the essential norm of a weighted composition operator relative to the class of Dunford-Pettis operators or the class of weakly compact operators, on the space ${\mathscr{H}}^{\infty }$ of Dirichlet series. As particular cases, we obtain the precise value of the generalized essential norm of a composition operator and of a multiplication operator.

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.

On présente dans cet exposé une approche semi-classique déduite des résultats de N. Burq, P. Gérard et N. Tzvetkov [4] permettant de démontrer des inégalités de Strichartz pour un problème non captif. On retrouve ainsi des résultats de G. Staffilani et D. Tataru [16] (obtenus pour une perturbation de la métrique à support compact). On donne aussi des généralisations de ces résultats au cas d’une perturbation à longue portée

On étudie dans ce travail les projections de norme 1 du bidual $E^{\text{'}\text{'}}$ d’un espace de Banach $E$ sur l’image canonique $i_E\left(E\right)$ de $E$ dans $E^{\text{'}\text{'}}$. On montre que dans un certain nombre de cas, il y a unicité de la projection de norme 1. On en déduit des théorèmes d’existence et d’unicité du prédual de $E$. On donne ensuite diverses applications, en particulier aux espaces dont la norme est différentiable sur un ensemble dense et aux espaces ne contenant pas ${\ell }^1\left(N\right)$.