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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.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 47F05 Partial differential operators
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##### References:
 [1] Auscher, P.: On necessary and sufficient conditions for lp estimates of Riesz transform associated elliptic operators on rn and related estimates. Mem. amer. Math. soc. 186, No. 871 (2007) · Zbl 1221.42022 [2] Auscher, P.; Martell, J. M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights · Zbl 1213.42030 [3] Auscher, P.; Martell, J. M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type · Zbl 1210.42023 [4] Auscher, P.; Tchamitchian, Ph.: Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux). Ann. inst. Fourier 45, 721-778 (1995) [5] Auscher, P.; Tchamitchian, Ph.: Square root problem for divergence operators and related topics. Astérisque 249 (1998) · Zbl 0909.35001 [6] Auscher, P.; Coulhon, T.; Duong, X. T.; Hofmann, S.: Riesz transforms on manifolds and heat kernel regularity. Ann. sci. École norm. Sup. (4) 37, No. 6, 911-957 (2004) · Zbl 1086.58013 [7] Auscher, P.; Hofmann, S.; Lacey, M.; Mcintosh, A.; Tchamitchian, Ph.: The solution of the Kato square root problem for second order elliptic operators on rn. Ann. of math. (2) 156, 633-654 (2002) · Zbl 1128.35316 [8] Blunck, S.; Kunstmann, P.: Weighted norm estimates and maximal regularity. Adv. differential equations 7, No. 12, 1513-1532 (2002) · Zbl 1045.34031 [9] Blunck, S.; Kunstmann, P.: Calderón -- Zygmund theory for non-integral operators and the H$\infty$-functional calculus. Rev. mat. Iberoamericana 19, No. 3, 919-942 (2003) · Zbl 1057.42010 [10] Duong, X. T.; Yan, L.: Commutators of BMO functions and singular integral operators with non-smooth kernels. Bull. austral. Math. soc. 67, No. 2, 187-200 (2003) · Zbl 1023.42010 [11] Franchi, B.; Pérez, C.; Wheeden, R.: Self-improving properties of John -- Nirenberg and Poincaré inequalities on spaces of homogeneous type. J. funct. Anal. 153, No. 1, 108-146 (1998) · Zbl 0892.43005 [12] García-Cuerva, J.; De Francia, J. L. Rubio: Weighted norm inequalities and related topics. North-holland math. Stud. 116 (1985) · Zbl 0578.46046 [13] Grafakos, L.: Classical and modern Fourier analysis. (2004) · Zbl 1148.42001 [14] Kalton, N.; Weis, L.: The H$\infty$-calculus and sums of closed operators. Math. ann. 321, 319-345 (2001) · Zbl 0992.47005 [15] Le Merdy, C.: On square functions associated to sectorial operators. Bull. soc. Math. France 132, No. 1, 137-156 (2004) · Zbl 1066.47013 [16] Martell, J. M.: Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Studia math. 161, 113-145 (2004) · Zbl 1044.42019 [17] Mcintosh, A.: Operators which have an H$\infty$ functional calculus. Center for math. And appl. 14, 210-231 (1986) [18] Pérez, C.; Trujillo-Gonzalez, R.: Sharp weighted estimates for multilinear commutators. J. London math. Soc. (2) 65, 672-692 (2002) · Zbl 1012.42008 [19] Shen, Z.: Bounds of Riesz transforms on lp spaces for second order elliptic operators. Ann. inst. Fourier 55, No. 1, 173-197 (2005) · Zbl 1068.47058 [20] Stein, E. M.: Singular integrals and differentiability of functions. (1970) · Zbl 0207.13501 [21] Strömberg, J. O.; Torchinsky, A.: Weighted Hardy spaces. Lecture notes in math. 1381 (1989) · Zbl 0676.42021 [22] Weis, L.: Operator-valued Fourier multiplier theorems and maximal lp-regularity. Math. ann. 319, 735-758 (2001) · Zbl 0989.47025