Weak type (p,q)-inequalities for the Haar system and differentially subordinated martingales
✍ Scribed by Adam Oseçkowski
- Publisher
- John Wiley and Sons
- Year
- 2012
- Tongue
- English
- Weight
- 205 KB
- Volume
- 285
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
For any \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$1\leq p,,q<\infty$\end{document}, we determine the optimal constant \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$C_{p,q}$\end{document} such that the following holds. If \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(h_k)_{k\geq 0}$\end{document} is the Haar system on [0,1], then for any vectors a~k~ from a separable Hilbert space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal{H}$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\varepsilon_k\in {-1,1}$\end{document}, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$k=0,,1,,2,\ldots,$\end{document} we have
This is generalized to the sharp weak‐type inequality
where X, Y stand for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal{H}$\end{document}‐valued martingales such that Y is differentially subordinate to X.