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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 $\varphi(L)$ for bounded holomorphic functions $\varphi$, 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 in Part I [loc. cit.] and the off-diagonal estimates in Part II [loc. cit.]. We also obtain commutator results with BMO functions.

42B20Singular and oscillatory integrals, several variables
47F05Partial differential operators
Full Text: DOI
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