In this paper we prove boundedness results on atomic Hardy type spaces for multipliers of the spherical transform on noncompact symmetric spaces of arbitrary rank. The multipliers we consider satisfy either inhomogeneous or homogeneous Mihlin-Hörmander type conditions. In particular, we are able to treat the case of {strongly singular multipliers} whose convolution kernels are not integrable at infinity. Thus our results apply also to negative and imaginary powers of the Laplacian.

Work partially supported by PRIN 2015 "Real and complex manifolds: geometry, topology and harmonic analysis". The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)