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

Let $L_{\nu }=-{\mathrm{\partial }}_x^2-(\nu -1)x^{-1}{\mathrm{\partial }}_x$ be the Bessel operator on the half-line $X_{\nu }=[0,\mathrm{\infty })$ with measure $x^{\nu -1}\mathrm{d}x$. In this work we study singular integral operators associated with the Laplacian ${\mathrm{\Delta }}_{\nu }=-{\mathrm{\partial }}_u^2+e^{2u}{L}_{\nu }$ on the product $G_{\nu }$ of $X_{\nu }$ and the real line with measure $\mathrm{d}u$. For any $\nu \ge 1$, the Laplacian ${\mathrm{\Delta }}_{\nu }$ 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 ${\mathrm{\Delta }}_{\nu }$, the metric-measure space $G_{\nu }$ has exponential volume growth. We prove a sharp $L^p$ spectral multiplier theorem of Mihlin–Hörmander type for ${\mathrm{\Delta }}_{\nu }$, as well as the $L^p$-boundedness for $p\in (1,\mathrm{\infty })$ 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 $G_{\nu }$, and establish large-time gradient heat kernel estimates for ${\mathrm{\Delta }}_{\nu }$. 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.