Endpoint results for spherical multipliers on noncompact symmetric spaces

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–Hormander type condi- tions. 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.