0$ such that $|\phi(z)| \leq c(|z|/(1+|z|^2))$ and $\phi(x) > 0$ for positive real $x$. Put $\phi_k(\lambda)=\phi(2^{-k}\lambda)$. We define a vector valued operator $S_{\phi}$ by the formula: $$ S_{\phi}(f) = \{\phi_k(L)f\}_{k=-\infty}^{\infty}.$$ \begin{fact} $S_{\phi}$ is bounded from $L^p(dx)$ to $L^p(\ell^2)$. \end{fact} {\it Proof}: This is a consequence of the holomorphic multiplier theorem from \cite{C:m} or \cite{S:t}, using classical arguments. \qed Choose $\psi \in C_c^{\infty}({\cal R}_+)$ such that $\sum_k \psi(2^kx) = 1$ for all $x>0$. Let $m_k(\lambda)=\psi(2^{-k}\lambda)m(\lambda)$ and $h_k=\phi_k^{-2}m_k$. Then $$ m(L) = \sum m_k(L) = \sum \phi_k(L)h_k(L)\phi_k(L) = S_{\phi}^{*}HS_{\phi}$$ where $H$ is the bounded operator on $L^2(\ell^ 2)$ given by the formula: $$ H\vfk=\{h_k(L)f_k\}_{k=-\infty}^{\infty}$$ and $S_{\phi}^{*}:L^2(\ell^ 2) \mapsto L^2(dx)$ is the adjoint of $S_{\phi}$. Thus, to prove Theorem \ref{main-t} we only need to prove that $H$ is bounded on $L^p(dx, \ell^2)$. % Of course $H$ is bounded on $L^2(dx, \ell^2)$. \begin{lem}\label{vec-mult1} There exists $C$ such that for all $k$ and $f_k$ $$ \|h_k(L)\|_{L^1,L^1} \leq C$$ $$ |h_k(L)f_k|(x) \leq C \sup_{t>0}\exp(-tL)|f_k|(x)$$ \end{lem} {\it Proof}: For $x\in {\cal R}^n$ put $\eta_k(x) = h_k(2^k|x|^2)$. The functions $\eta_k$ are in $C_c^{n+1}$ with uniform bounds on their support and their derivatives so $$ |\widehat{\eta_k}|(y) \leq C_1 (1+|y|)^{-n-1} $$ where $\widehat{\phantom{X}}$ denotes the Fourier transform and $C_1$ does not depend on $k$. Next, there is a nonnegative integrable function $w$ such that $$ C_1(1+|y|)^{-n-1} \leq \int_0^{\infty}w(t)\widehat {e_t}(y)dt$$ where $e_t(x) = \exp(-t|x|^2)$. For example, we can take multiple of $(1 + t)^{-3/2}$ as $w$. Consequently, $$ |h_k(-\Delta)|(x) \leq C_1\int_0^{\infty}w(t)\exp(t2^{-k}\Delta)(x)dt$$ where $\Delta$ is laplacian on ${\cal R}^n$. By \cite{H:s} the last inequality remains valid on $AN$: $$ |h_k(L)|(x) \leq C_1\int_0^{\infty}w(t)\exp(-t2^{-k}L)(x)dt$$ Since $\|\exp(t2^{-k}L)\|_{L^1,L^1} = 1$ the first claim follows. To get the second claim we note that $$ |h_k(L)f_k|(x) \leq C_1\int_0^{\infty}w(t)\exp(-t2^{-k}L)|f_k|(x)dt$$ $$ \leq C_1\int_0^{\infty} w(t)dt\sup_{t>0}\exp(-t2^{-k}L)|f_k|(x) = C_2\sup_{t>0}\exp(-tL)|f_k|(x).$$ \qed Now $$ \sup_k |h_k(L)f_k|(x) \leq C \sup_{t>0}\exp(-tL) (\sup_k |f_k|)(x)$$ Since the semigroup maximal function is bounded on $L^p$, $H$ is bounded on $L^p(dx, \ell^{\infty})$. Next, since $$\|h_k(L)\|_{L^1(dx),L^1(dx)} \leq C$$ we have $$ \|H\vfk\|_{L^1(dx,\ell^1)} = \|\sum_k|h_k(L)f_k|\|_{L^1(dx)} =\sum_k\|h_k(L)f_k\|_{L^1(dx)}$$ $$ \leq C \sum_k\|f_k\|_{L^1(dx)} = \|\vfk\|_{L^1(dx,\ell^1)}.$$ By analytic interpolation between $L^1(dx,\ell^1)$ and $L^2(dx,\ell^2)$ $H$ is bounded on $L^p(dx,\ell^p)$, $1\leq p \le 2$. Again, by interpolation between $L^p(dx,\ell^p)$ and $L^p(dx,\ell^{\infty})$ $H$ is bounded on $L^p(dx,\ell^2)$, $1\leq p \le 2$. We handle $p > 2$ by duality, which ends the proof. \section{Possible improvements and limitations} In Lemma \ref{vec-mult1} it is enough to bound the Sobolev norm $H((n+1)/2 +\varepsilon)$ of $h_n$. Since this is the only place where we use regularity of $m$, the main theorem remain valid if $m$ only satisfies: $$ \sup_{t>0} \|\psi m(t\cdot)\|_{H(s)} < \infty. $$ with $s>(n+1)/2$. Lemma \ref{vec-mult1} (with $n$ replaced by apropriate values like in \cite{CGHM:m}) remains valid for distinguished laplacian on all (not necessarly complex) Iwasawa AN groups, however the proof is much more complicated. Since our argument is based on use of maximal function it probably cannot be improved to give expected critical exponent $n/2$. Also, it is probably impossible to get the weak type $(1, 1)$ of the multiplier operator preserving the simplicity of the argument. Finally, let us mention that the related problem of bounding Riesz transforms requires estimates of derivatives of the semigroup kernel, hence a quite different method. \begin{thebibliography}{00} \bibitem{C:m} {M. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117 (1983), 267--283. } \bibitem{CGHM:m} {M. Cowling, S. Giulini, A. Hulanicki, G. Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, {\it Studia Math.} 111 (1994), 103--121.} \bibitem{H:s} {W. Hebisch, The subalgebra of \hbox{$L^{1}(AN)$} generated by the laplacian, Proc. AMS 117 (1993), 547--549.} \bibitem{HSt} {W.~Hebisch, T.~ Steger, Multipliers and singular integrals on exponential growth groups, Math. Zeit. \textbf{245} (2003), 37--61.} \bibitem{S:t} {E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. of Math. Stud. 63, Princeton Univ. Press, Princeton 1970.} \end{thebibliography} Mathematical Institute, Wroc\l aw University, pl. Grunwaldzki 2/4, 50-384 Wroc\l aw, Poland E-mail: {\tt hebisch@math.uni.wroc.pl} \end{document}