Let \(G\) be a connected noncompact semisimple Lie group with finite center and maximal compact subgroup \(K\). If \(u\) is a \(K\)-bi-invariant distribution on \(G\) we denote by \(\widetilde u\) its spherical Fourier transform. This paper is devoted to the study of operators \(T_u\) of the form \(T_uf=f*u\), where \(u\) is a \(K\)-bi-invariant distribution on \(G\), \(f\) is in \(L^p(G/K)\) for some \(p\) in \((1,\infty)\), and convolution is on the group \(G\). The authors give sufficient conditions on \(\widetilde u\) so that \(T_u\) is bounded on \(L^p(G/K)\) for some \(p\) in \((1,\infty)\). Let \(n\) and \(\ell\) denote the dimension of \(G/K\) and its real rank, respectively. If \(p\) is in \((1,\infty)\), let \(\delta(p)\) denote the number \(2/p-1\). A classical condition says that if \(T_u\) is bounded on \(L^p(G/K)\), then \(\widetilde{u}\) extends to a bounded holomorphic function on a certain tube, which we denote \(\mathbb{T}_{\delta(p)}\), in \(\mathbb{C}^\ell\). Let \(\omega\) be a meromorphic function, which is holomorphic in a neighbourhood of \(\mathbb{T}_1\) and grows of order \((n-\ell)/2\) at infinity. Finally, let \(Y\) be a Banach space which is continuously imbedded in the space of the \(L^p\) Euclidean Fourier multipliers on \(\mathbb{R}^\ell\) for all \(p\) in (1,2), and denote by \(Y_p\) the complex interpolation space \([Y,L^\infty(\mathbb{R}^\ell)^\theta]\), if \(\theta=2/p'\). The main result of the paper is as follows. Theorem. Let \(G\) be a connected noncompact semisimple Lie group which is either complex or rank one. Suppose that \(q\) is in (1,2), and let \(u\) be a \(K\)-bi-invariant distribution on \(G\) such that \(\widetilde u\) is a Weyl-invariant holomorphic function on \(\mathbb{T}_{\delta(q)}\). If \(\omega^{\delta(p)}\widetilde u\) is in \(Y_p\) uniformly in every proper subtube of \(\mathbb{T}_{\delta(q)}\), then \(T_u\) is bounded on \(L^p(G/K)\) for all \(p\) in \((q,q')\). -- The authors also show that the Clerc-Stein multiplier theorem is an easy corollary of this result. A similar result is proved for functions of the standard isotropic nearest neighbour Laplacian in the context of free groups.