Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

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.