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

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