Abstract
In this paper we introduce a new atomic Hardy space \(X^1(\gamma )\) adapted to the Gauss measure \(\gamma \), and prove the boundedness of the first order Riesz transform associated with the Ornstein–Uhlenbeck operator from \(X^1(\gamma )\) to \(L^1(\gamma )\). We also provide a new, short and almost self-contained proof of its weak-type (1, 1).
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Notes
Given a bounded operator T on \(L^2(\gamma )\), we say that a distribution \(K_T\) on \(\mathbb {R}^n \times \mathbb {R}^n\) is its Schwartz kernel with respect to the Lebesgue measure if
$$\begin{aligned} Tf(x)= \int _{\mathbb {R}^n} K_T(x,y) f(y)\, \mathrm {d}y \end{aligned}$$for a.e. \(x\in \mathbb {R}^n\).
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Acknowledgements
It is a great pleasure to thank Giancarlo Mauceri and Stefano Meda for several fruitful discussions and their constant help and support. I also thank the anonymous referees for valuable comments and remarks.
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Communicated by Krzysztof Stempak.
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Bruno, T. Endpoint Results for the Riesz Transform of the Ornstein–Uhlenbeck Operator. J Fourier Anal Appl 25, 1609–1631 (2019). https://doi.org/10.1007/s00041-018-09648-8
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DOI: https://doi.org/10.1007/s00041-018-09648-8