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

Let Lν = −∂ 2 x − (ν − 1)x−1∂x be the Bessel operator on the halfline Xν = [0, ∞) with measure x ν−1 dx. In this work we study singular integral operators associated with the Laplacian ∆ν = −∂ 2 u +e 2uLν on the product Gν of Xν and the real line with measure du. For any ν ≥ 1, the Laplacian ∆ν is left-invariant with respect to a noncommutative hypergroup structure on Gν, which can be thought of as a fractional-dimension counterpart to ax+b groups. In particular, equipped with the Riemannian distance associated with ∆ν, the metric-measure space Gν has exponential volume growth. We prove a sharp Lp spectral multiplier theorem of Mihlin–H¨ormander type for ∆ν, as well as the Lp-boundedness for p ∈ (1, ∞) 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 Gν, and establish large-time gradient heat kernel estimates for ∆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