Multivariate H\"ormander-type multiplier theorem for the Hankel transform

Abstract

Let H(f)(x)=\int_{(0,infty)^d} f(v) E_{x}(v) d\nu(v), be the multivariable Hankel transform, where E_{x}(v)=\prod_{k=1}^d (x_k v_k)^{-a_k+1/2} J_{a_k-1/2}(x_k v_k), d\nu(v)=v^a dv, a=(a_1,...,a_d). We give sufficient conditions on a bounded continuous function m(v) which guarantee that the operator H(m Hf) is bounded on L^p(d\nu) and of weak-type (1,1), or bounded on the Hardy space H^1((0,infty)^d, d\nu) in the sense of Coifman-Weiss.