This is an announcement for the paper "Absolutely summing operators and atomic decomposition in bi-parameter Hardy spaces" by Paul F.X. Muller and Johanna Penteker.

Abstract: For $f \in H^p(\delta^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator [\mathcal{M}_f:\ell^{\infty} \rightarrow H^p(\delta^2), \quad (\varphi_{I\times J}) \mapsto \sum \varphi_{I\times J}f_{I \times J}h_{I \times J}. ] Our method yields a constructive proof of Pisier's decomposition of $f \in H^p(\delta^2)$ [|f|=|x|^{1-\theta}|y|^{\theta}\quad\quad \text{ and }\quad\quad |x|_{X_0}^{1-\theta}|y|^{\theta}_{H^2(\delta^2)}\leq C|f|_{H^p(\delta^2)}, ] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(\delta^2)$ and $H^2(\delta^2)$. Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_f$ involves the Haar coefficients of $f$ and its atomic decomposition. We treated the one-parameter $H^p$-spaces in [P.F.X M"uller, J.Penteker, $p$-summing multiplication operators, dyadic Hardy spaces and atomic decomposition, Houston Journal Math.,41(2):639-668,2015.].

Archive classification: math.FA

Mathematics Subject Classification: 42B30 46B25 46B09 46B42 46E40 47B10 60G42

Remarks: 10 pages

Submitted from: johanna.penteker@jku.at

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1512.04790

or

http://arXiv.org/abs/1512.04790