2 8 M ar 2 00 6 WEIGHTED NORM INEQUALITIES , OFF - DIAGONAL ESTIMATES AND 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 inequalities for singular “non-integral” operators arising from L ; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−∆)1/2L−1/2) and its inverse L1/2(−∆)−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal L-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.