Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator
Denote by \(\gamma\) the Gaussian measure on \({\mathbb R}^d\). Let \({\mathcal L}\) be the Ornstein--Uhlenbeck operator. Its spectral resolution is \[ {\mathcal L} = \sum_{j=0}^{\infty} j {\mathcal P}_j, \] where \({\mathcal P}_j\) denotes the orthogonal projection onto the linear Hermite polynomials of degree \(j\) in \(d\) variables. Consider the sequence \(M:{\mathbb N} \to {\mathbb C}\), defined by \[ M(j) = \begin{cases} 0 & \text{if \quad } j=0, \\ j^{-1/2} & \text{if \quad } j=1,2, \ldots .\\ \end{cases} \] The first-order Riesz transforms associated with \({\mathcal L}\) are defined by \[ R_i := \partial_i M({\mathcal L}) \quad \text{and} \quad S_i := M({\mathcal L}) \partial_i \quad \text{where}\quad i= 1, \ldots d. \] \textit{P. A. Meyer} [Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 179--193 (1984; Zbl 0543.60078)] proved that \(R_i\) and \(S_i\) are bounded on \(L^p(\gamma)\) where \(1<p<\infty\). \textit{G. Mauceri} and \textit{S. Meda} [J. Funct. Anal. 252, No. 1, 278--313 (2007; Zbl 1136.46027)] proved that \(R_i\) are bounded from \(L^{\infty}(\gamma)\) to \(BMO(\gamma)\). In dimension one, the authors prove that \(R_1\) and \(S_1\) are bounded from \(H^1(\gamma)\) to \(L^1(\gamma)\). Surprisingly, in higher dimensions, they prove the following: when \(d \geq 2\), \(R_i\) and \(S_i\) are bounded from \(L^{\infty}(\gamma)\) to \(BMO(\gamma)\), but not bounded from \(H^1(\gamma)\) to \(L^1(\gamma)\).
scientific article