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Weighted norm inequalities, off-diagonal estimates and elliptic operators. III: Harmonic analysis of elliptic operators. (English) Zbl 1213.42029
Summary: This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators [for Parts I, II and IV, see, respectively, ibid. 212, No. 1, 225–276 (2007; Zbl 1213.42030); J. Evol. Equ. 7, No. 2, 265–316 (2007; Zbl 1210.42023); and Math. Z. 260, No. 3, 527–539 (2008; Zbl 1214.58010)]. For $L$ in some class of elliptic operators, we study weighted ${L}^{p}$ norm inequalities for singular “non-integral” operators arising from $L$; those are the operators $\phi \left(L\right)$ for bounded holomorphic functions $\phi$, the Riesz transforms $\nabla {L}^{-1/2}$ (or ${\left(-{\Delta }\right)}^{1/2}{L}^{-1/2}$) and its inverse ${L}^{1/2}{\left(-{\Delta }\right)}^{1/2}$, 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 in Part I [loc. cit.] and the off-diagonal estimates in Part II [loc. cit.]. We also obtain commutator results with BMO functions.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 47F05 Partial differential operators