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Hardy Spaces and Differentiation of the Integral in the Product Setting

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Hardy Spaces and Differentiation of the Integral in the Product Setting

Cabral, Raquel de Montalvão (2014) Hardy Spaces and Differentiation of the Integral in the Product Setting. PhD thesis, Concordia University.

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Abstract

This work concerns strong differentiation and operators on product Hardy spaces. We show, by counterexample, that strong differentiability of the integral fails even for functions in the intersection of H_{rect}^{1}(\mathbb{R}×\mathbb{R}) with L(log⁡L)^{ε}(\mathbb{R}^{2}) for all 0<ε<1. Our example is a modification of a function that appears in a work of J. M. Marstrand, where he makes a claim concerning” approximately independent sets”. We generalize his claim and, as a corollary, we obtain a version of the second Borel-Cantelli Lemma. In addition, we prove that a function f created by Papoulis to show that the strong differentiability of ∫f does not imply the same behavior for ∫|f|, belongs to the product Hardy space H_{rect}^{p}(\mathbb{R}×\mathbb{R}). The method that we develop to approach this example allows us to relax the sufficient conditions of the Chang-Fefferman atomic decomposition. In analogy with the proof of this result, we demonstrate that a theorem of R. Fefferman, which concludes H^{p}→L^{p}, 0<p≤1, boundedness of two-parameter operators from their behavior on rectangle atoms, can be generalized to settings with more parameters. This generalization enables us to extend a theorem of Pipher concerning boundedness of multiparameter Calderón-Zygmund operators from H^{p} to L^{p}. Furthermore, we present variants of Journé's Lemma, two of which hold for the product of \mathbb{R} with a metric measure space satisfying certain conditions.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (PhD)
Authors:Cabral, Raquel de Montalvão
Institution:Concordia University
Degree Name:Ph. D.
Program:Mathematics
Date:25 August 2014
Thesis Supervisor(s):Dafni, Galia Devora
ID Code:978954
Deposited By: RAQUEL CABRAL
Deposited On:26 Nov 2014 14:26
Last Modified:18 Jan 2018 17:48
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