×

Multilinear version of reversed Hölder inequality and its applications to multilinear Calderón-Zygmund operators. (English) Zbl 1263.47012

The authors provide a generalized reverse Hölder inequality stating that, if \(\omega_{i}\in A_\infty\), then for every cube \(Q\), \[ \int_Q\prod _{i =1}^m \omega_i^{\theta_i} \geq \prod_{i = 1}^m \left (\frac {\int_Q \omega_i}{[\omega_i]_{A_\infty}} \right)^{\theta_i}, \] where \(\sum_{i=1}^m\theta_{i}=1\), \(0\leq \theta_i\leq 1\). They present an extension of the reverse Jensen inequality in the theory of weighted inequalities and investigate some problems arising in the study of multilinear Calderón-Zygmund operators (and maximal operators) with multiple weights.

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc., 340 (1993), 253-272. · Zbl 0795.42011 · doi:10.2307/2154555
[2] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. · Zbl 0291.44007
[3] D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc., 347 (1995), 2941-2960. · Zbl 0851.42016 · doi:10.2307/2154763
[4] J. García-Cuerva, Weighted \(H^p\) spaces, Dissertation Math., 162 (1979), 1-63.
[5] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Studies, 116 , North-Holland, Amsterdam, 1985. · Zbl 0578.46046
[6] L. Grafakos, Classic and Modern Fourier Analysis, Prentice Hall, New Jersey, 2004. · Zbl 1148.42001
[7] L. Grafakos and N. Kalton, Multilinear Calderón-Zygmund operators on Hardy spaces, Collect. Math., 52 (2001), 169-179. · Zbl 0986.42008
[8] L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164. · Zbl 1032.42020 · doi:10.1006/aima.2001.2028
[9] L. Grafakos and R. H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J., 51 (2002), 1261-1276. · Zbl 1033.42010 · doi:10.1512/iumj.2002.51.2114
[10] A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math., 220 (2009), 1222-1264. · Zbl 1160.42009 · doi:10.1016/j.aim.2008.10.014
[11] W. Li, Q. Xue and K. Yabuta, Multilinear Calderón-Zygmund operators on weighted Hardy spaces, Studia Math., 199 (2010), 1-16. · Zbl 1201.42009 · doi:10.4064/sm199-1-1
[12] W. Li, Q. Xue and K. Yabuta, Maximal operator for multilinear Calderón-Zygmund singular integral operators on weighted Hardy spaces, J. Math. Anal. Appl., 373 (2011), 384-392. · Zbl 1204.42024 · doi:10.1016/j.jmaa.2010.07.057
[13] J.-O. Stromberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381 , Springer-Verlag, Berlin, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.