Singular integrals on $ax+b$ hypergroups and an operator-valued spectral multiplier theorem

Abstract

Let $L_ν= -\partial_x^2-(ν-1)x^{-1} \partial_x$ be the Bessel operator on the half-line $X_ν= [0,\infty)$ with measure $x^{ν-1} \,\mathrm{d} x$. In this work we study singular integral operators associated with the Laplacian $Δ_ν= -\partial_u^2 + e^{2u} L_ν$ on the product $G_ν$ of $X_ν$ and the real line with measure $\mathrm{d} u$. For any $ν\geq 1$, the Laplacian $Δ_ν$ is left-invariant with respect to a noncommutative hypergroup structure on $G_ν$, which can be thought of as a fractional-dimension counterpart to $ax+b$ groups. In particular, equipped with the Riemannian distance associated with $Δ_ν$, the metric-measure space $G_ν$ has exponential volume growth. We prove a sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type for $Δ_ν$, as well as the $L^p$-boundedness for $p \in (1,\infty)$ of the associated first-order Riesz transforms. To this purpose, we develop a Calderón--Zygmund theory à la Hebisch--Steger adapted to the nondoubling structure of $G_ν$, and establish large-time gradient heat kernel estimates for $Δ_ν$. In addition, the Riesz transform bounds for $p > 2$ hinge on an operator-valued spectral multiplier theorem, which we prove in greater generality and may be of independent interest.