Let $\nu \in [-1/2, \infty)^n$, $n \geq 1$, and let $\mathcal{L}_\nu$ be a self-adjoint extension of the differential operator

\begin{equation} L_\nu := \sum_{i=1}^n \left[ - \frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2} (\nu_i^2 - \frac{1}{4}) \right] \end{equation}

on $C_c^\infty(\mathbb{R}^n_+)$ as the natural domain. In this talk, we will discuss the boundedness of the Riesz transforms associated with $\mathcal{L}_\nu$. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with $\mathcal{L}_\nu$ and prove that the Riesz transforms are bounded on these Hardy spaces and Campanato spaces. This completes the description of the boundedness of the Riesz transform in the Laguerre expansion setting.

The video of this talk is available on the IISc Math Department channel.