Dual spaces of mixed-norm martingale Hardy spaces

Ferenc Weisz

doi:10.3934/cpaa.2020285

Article Contents

2021, Volume 20, Issue 2: 681-695. Doi: 10.3934/cpaa.2020285

This issue Previous Article Random data theory for the cubic fourth-order nonlinear Schrödinger equation Next Article Elliptic problems with rough boundary data in generalized Sobolev spaces

Dual spaces of mixed-norm martingale Hardy spaces

Ferenc Weisz

Department of Numerical Analysis, Eötvös L. University, H-1117 Budapest, Pázmány P. sétány 1/C., Hungary

Received: April 30, 2020

Revised: September 30, 2020

Early access: December 2020

Published: February 2021

This research was supported by the Hungarian National Research, Development and Innovation Office-NKFIH, KH130426

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Abstract

In this paper, we generalize the Doob's maximal inequality for mixed-norm $ L_{\vec{p}} $ spaces. We consider martingale Hardy spaces defined with the help of mixed $ L_{{\vec{p}}} $-norm. A new atomic decomposition is given for these spaces via simple atoms. The dual spaces of the mixed-norm martingale Hardy spaces is given as the mixed-norm $ BMO_{\vec{r}}(\vec{\alpha}) $ spaces. This implies the John-Nirenberg inequality $ BMO_{1}(\vec{\alpha}) \sim BMO_{\vec{r}}(\vec{\alpha}) $ for $ 1<\vec{r}<\infty $. These results generalize the well known classical results for constant $ p $ and $ r $.

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Mathematics Subject Classification: Primary: 42B35; Secondary: 42B30, 60G42, 42B25, 46E30.

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