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

Let $L_\nu = -\partial_x^2-(\nu-1)x^{-1} \partial_x$ be the Bessel operator on the half-line $X_\nu = [0,\infty)$ with measure $x^{\nu-1} \,\mathrm{d} x$. In this work we study singular integral operators associated with the Laplacian $\Delta_\nu = -\partial_u^2 + e^{2u} L_\nu$ on the product $G_\nu$ of $X_\nu$ and the real line with measure $\mathrm{d} u$. For any $\nu \geq 1$, the Laplacian $\Delta_\nu$ is left-invariant with respect to a noncommutative hypergroup structure on $G_\nu$, which can be thought of as a fractional-dimension counterpart to $ax+b$ groups. In particular, equipped with the Riemannian distance associated with $\Delta_\nu$, the metric-measure space $G_\nu$ has exponential volume growth. We prove a sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander type for $\Delta_\nu$, 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 Calder\'on--Zygmund theory \`a la Hebisch--Steger adapted to the nondoubling structure of $G_\nu$, and establish large-time gradient heat kernel estimates for $\Delta_\nu$. 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.