In this paper, we give theoretical results on Macaev ideal and Dixmier trace. Then we give a characterization of antiholomorphic symbols \overline f such that the Hankel operator {H_{\overline f }} on a Bergman weighted space is in an ideal of Macaev and we give the Dixmier trace. For this, we look at the behavior of Schatten’s norms \mathcal{S}^p when p tends to 1, using results of Engliš and Rochberg on Bergman space. We also give results on powers of such operators., Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes \overline f tels que l’opérateur de Hankel {H_{\overline f }} sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten \mathcal{S}^p quand p tend vers 1 et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs., Romaric Tytgat., and Obsahuje seznam literatury
For any holomorphic function f on the unit polydisk D n we consider its restriction to the diagonal, i.e., the function in the unit disc D ⊂ C defined by Diag f(z) = f(z, . . . , z), and prove that the diagonal map Diag maps the space Qp,q,s(D n ) of the polydisk onto the space Qbq p,s,n(D ) of the unit disk.
We develop a theory of removable singularities for the weighted Bergman space ${\mathcal A}^p_\mu (\Omega )=\lbrace f \text{analytic} \text{in} \Omega \: \int _\Omega |f|^p \mathrm{d}\mu < \infty \rbrace $, where $\mu $ is a Radon measure on $\mathbb{C}$. The set $A$ is weakly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) \subset \text{Hol}(\Omega )$, and strongly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) = {\mathcal A}^p_\mu (\Omega )$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop {\mathrm BMO}$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if $\mu $ is absolutely continuous with respect to the Lebesgue measure $m$, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When $\mathrm{d}\mu = w\mathrm{d}m$ and $w$ is a Muckenhoupt $A_p$ weight, $1<p<\infty $, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p^{\prime }=p/(p-1)$ and the dual weight $w^{\prime }=w^{1/(1-p)}$.