Hardy spaces, Campanato spaces and higher order Riesz transforms associated with Bessel operators
The Anh Bui
TL;DR
The paper develops a comprehensive harmonic‐analysis framework for the multivariate Bessel operator $Δ_ν$ by defining Hardy spaces $H^p_{Δ_ν}$ and local Campanato/BMO spaces $BMO^{s,M}_{ρ}$ tied to the critical function $ρ$. It establishes sharp heat-kernel and derivative bounds, proves that higher-order Riesz transforms $δ_ν^k Δ_ν^{-|k|/2}$ are Calderón–Zygmund operators, and proves their boundedness on the Hardy and Campanato spaces for the natural range of $p$ governed by $γ_ν=ν_{ ext{min}}+1/2$. A key contribution is the equivalence $H^p_{ρ}(ℝ^n_+) ≃ H^p_{Δ_ν}(ℝ^n_+)$ and the duality $(H^p_{Δ_ν})^* = BMO^{s,M}_{ρ}$ with $s=n(1/p-1)$, plus a detailed construction of the Campanato framework via a ρ–ball covering. These results extend Hardy/BMO theory to Bessel/Laguerre-type operators and provide a robust foundation for higher-order Riesz transforms in this non-Euclidean setting.
Abstract
Let $ν= (ν_1, \ldots, ν_n) \in (-1/2, \infty)^n$, with $n \ge 1$, and let $Δ_ν$ be the multivariate Bessel operator defined by \[ Δ_ν = -\sum_{j=1}^n\left( \frac{\partial^2}{\partial x_j^2} - \frac{ν_j^2 - 1/4}{x_j^2} \right). \] In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator $Δ_ν$. We then study the higher-order Riesz transforms associated with $Δ_ν$. First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with $Δ_ν$.