Hardy spaces and Campanato spaces associated with Laguerre expansions and higher order Riesz transforms

Abstract

Let \(\mathcal{L}_ν\) be the Laguerre differential operator which is the self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2} \left(ν_i^2 - \frac{1}{4} \right) \right] \] initially defined on \(C_c^\infty(\mathbb{R}_+^n)\) as its natural domain, where \(ν\in [-1/2,\infty)^n\), \(n \geq 1\). In this paper, we first develop the theory of Hardy spaces \(H^p_{\mathcal{L}_ν}\) associated with \(\mathcal{L}_ν\) for the full range \(p \in (0,1]\). Then we investigate the corresponding BMO-type spaces and establish that they coincide with the dual spaces of \(H^p_{\mathcal{L}_ν}\). Finally, we show boundedness of higher-order Riesz transforms on Lebesgue spaces, as well as on our new Hardy and BMO-type spaces.

44 pages