  Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$
is a space of homogeneous type. Assume that $L$ generates a holomorphic
semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy generalized $m$-th order
Gaussian estimates. In this article, we study singular and dyadically supported
spectral multipliers for abstract self-adjoint operators. We show that in this
setting sharp spectral multiplier results follow from Plancherel or Stein-Tomas
type estimates. These results are applicable to spectral multipliers for large
classes of operators including $m$-th order elliptic differential operators
with constant coefficients, biharmonic operators with rough potentials and
Laplace type operators acting on fractals.
