Endpoint estimates for higher order Gaussian Riesz transforms
Fabio Berra, Estefanía Dalmasso, Roberto Scotto
TL;DR
The paper analyzes endpoint bounds for higher-order Gaussian Riesz transforms associated with the Ornstein--Uhlenbeck operator under the Gaussian measure. It introduces a new chain of Hardy-type spaces $X^k(\gamma)$, each containing $X^{k+1}(\gamma)\subsetneq X^k(\gamma)\subsetneq \mathcal{H}^1(\gamma)$, and proves that the old transforms $R_\alpha$ are bounded from $X^k(\gamma)$ to $L^1(\gamma)$ when $|\alpha|=k$, while the new transforms $R_\alpha^*$ are bounded from $\mathcal{H}^1(\gamma)$ to $L^1(\gamma)$ for all orders and dimensions. The approach combines a refined spectral calculus for $\mathcal{L}$, precise kernel and derivative estimates for $\mathcal{L}^{k/2}$, and atomic decompositions to control endpoint behavior; crucial tools include support-preserving properties of $\mathcal{L}^{-k}$ on atoms and off-diagonal decay for odd powers. These results extend Bruno's first-order endpoint bound to higher orders and establish a comprehensive endpoint theory for Gaussian Riesz transforms in the Ornstein--Uhlenbeck setting, with potential implications for Gaussian harmonic analysis and related PDEs.
Abstract
We will show that, contrary to the behavior of the higher order Riesz transforms studied so far on the atomic Hardy space $\mathcal{H}^1(\mathbb R^n, γ)$, associated with the Ornstein-Uhlenbeck operator with respect to the $n$-dimensional Gaussian measure $γ$, the new Gaussian Riesz transforms are bounded from $\mathcal{H}^1(\mathbb R^n, γ)$ to $L^1(\mathbb R^n, γ)$, for any order and dimension $n$. We will also prove that the classical Gaussian Riesz transforms of higher order are bounded from an adequate subspace of $\mathcal{H}^1(\mathbb R^n, γ)$ into $L^1(\mathbb R^n, γ)$, extending Bruno's result (J. Fourier Anal. Appl. 25, 4 (2019), 1609--1631) for the first order case.