Spectral multipliers for Laplacians associated to some Dirichlet forms

Let $\cLphi =-\Delta-\vp^{-1}\nabla\vp\cdot\nabla$ be the self-adjoint operator associated to the Dirichlet form $Q^\vp(f)=\int_{\BR^d} |\nabla f(x)|^2 \wrt\laphi(x)$, where $\vp$ is a positive $C^2$ function, $\wrt \laphi = \vp \wrt\la$ and $\la$ denotes Lebesgue measure on $\BR^d$. We study the boundedness on $L^p(\laphi)$ of spectral multipliers of $\cLphi $. We prove that if $\vp$ grows or decays at most exponentially at infinity and satisfies a suitable ``curvature condition", then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin type conditions at infinity are spectral multipliers of $L^p(\laphi)$. The parabolic region depends on~$\vp$, on~$p$, and on the infimum of the essential spectrum of the operator $\cLphi$ on~$\ld{\laphi}$. The sector depends on the angle of holomorphy of the semigroup generated by~$\cLphi$ on~$\lp{\laphi}$.