  This is the third part of a series of four articles on weighted norm
inequalities, off-diagonal estimates and elliptic operators. For $L$ in some
class of elliptic operators, we study weighted norm $L^p$ inequalities for
singular 'non-integral' operators arising from $L$ ; those are the operators
$\phi(L)$ for bounded holomorphic functions $\phi$, the Riesz transforms
$\nabla L^{-1/2}$ (or $(-\Delta)^{1/2}L^{-1/2}$) and its inverse
$L^{1/2}(-\Delta)^{-1/2}$, some quadratic functionals $g\_{L}$ and $G\_{L}$ of
Littlewood-Paley-Stein type and also some vector-valued inequalities such as
the ones involved for maximal $L^p$-regularity. For each, we obtain sharp or
nearly sharp ranges of $p$ using the general theory for boundedness of Part I
and the off-diagonal estimates of Part II. We also obtain commutator results
with BMO functions.
