Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Absolutely summing operators and atomic decomposition in bi-parameter Hardy spaces


Authors: Paul F. X. Müller and Johanna Penteker
Journal: Proc. Amer. Math. Soc. 145 (2017), 1221-1230
MSC (2010): Primary 42B30, 46B25, 46B09, 46B42, 46E40, 47B10, 60G42
DOI: https://doi.org/10.1090/proc/13300
Published electronically: November 3, 2016
MathSciNet review: 3589321
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle \mathcal {M}_f:\ell ^{\infty } \rightarrow H^p(\delta ^2), \quad ... ...hi _{I\times J}) \mapsto \sum \varphi _{I\times J}f_{I \times J}h_{I \times J}.$

Our method yields a constructive proof of Pisier's decomposition of $ f \in H^p(\delta ^2)$

$\displaystyle \vert f\vert=\vert x\vert^{1-\theta }\vert y\vert^{\theta }$$\displaystyle \quad \quad \text { and }\quad \quad \Vert x\Vert _{X_0}^{1-\theta }\Vert y\Vert^{\theta }_{H^2(\delta ^2)}\leq C\Vert f\Vert _{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 Houston Journal Math. 41 (2015), 639-668.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B30, 46B25, 46B09, 46B42, 46E40, 47B10, 60G42

Retrieve articles in all journals with MSC (2010): 42B30, 46B25, 46B09, 46B42, 46E40, 47B10, 60G42


Additional Information

Paul F. X. Müller
Affiliation: Institute of Analysis, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Austria
Email: paul.mueller@jku.at

Johanna Penteker
Affiliation: Institute of Analysis, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Austria
Email: johanna.penteker@gmail.com

DOI: https://doi.org/10.1090/proc/13300
Received by editor(s): December 17, 2015
Received by editor(s) in revised form: May 19, 2016
Published electronically: November 3, 2016
Additional Notes: This research was supported by the Austrian Science Foundation (FWF) Pr. Nr. P22549, Pr. Nr. P23987 and Pr. Nr. P28352
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society