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

Abstract

Let $L_{\hat{I}}½=-{\partial }_x^2-(\hat{I}½-1)x^{-1}{\partial }_x$ be the Bessel operator on the half-line $X_{\hat{I}}½=[0,\infty )$ with measure $x^{\hat{I}½-1}\mathrm{d}x$. In this work we study singular integral operators associated with the Laplacian $\hat{I}{”}_{\hat{I}}½=-{\partial }_u^2+e^{2u}{L}_{\hat{I}}½$ on the product $G_{\hat{I}}½$ of $X_{\hat{I}}½$ and the real line with measure $\mathrm{d}u$. For any $\hat{I}½\ge 1$, the Laplacian $\hat{I}{”}_{\hat{I}}½$ is left-invariant with respect to a noncommutative hypergroup structure on $G_{\hat{I}}½$, which can be thought of as a fractional-dimension counterpart to $ax+b$ groups. In particular, equipped with the Riemannian distance associated with $\hat{I}{”}_{\hat{I}}½$, the metric-measure space $G_{\hat{I}}½$ has exponential volume growth. We prove a sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type for $\hat{I}{”}_{\hat{I}}½$, 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 $G_{\hat{I}}½$, and establish large-time gradient heat kernel estimates for $\hat{I}{”}_{\hat{I}}½$. 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.

64 pages