Hardy spaces and Campanato spaces associated with Laguerre expansions and higher order Riesz transforms
The Anh Bui, Xuan Thinh Duong
TL;DR
This work develops a comprehensive Hardy-space theory for Laguerre expansions on \((0,\infty)^n\) for the full range \(p\in(0,1]\), including a Laguerre-adapted local atomic framework based on the critical function \(\rho_\nu\) and Campanato-type BMO duality with \(s=n(1/p-1)\). It proves the equivalence of Hardy spaces defined via atomic decompositions and those defined by heat semigroups and square functions, and it introduces Laguerre-specific tent-space tools to handle maximal characterizations. A key technical achievement is the sharp kernel estimates for the Laguerre heat kernel and its higher-order derivatives, enabling Calderón-Zygmund analysis of higher-order Riesz transforms \(\delta_\nu^k \mathcal{L}_\nu^{-|k|/2}\) for \(\nu\in[-1/2,\infty)^n\). The results establish boundedness of these transforms on both Hardy spaces \(H^p_{\rho_\nu}(\mathbb{R}^n_+)\) and Campanato spaces \(BMO^{s}_{\rho_\nu}(\mathbb{R}^n_+)\), enriching the harmonic analysis toolkit for Laguerre expansions and higher-order Riesz theory on noncompact domains.
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.