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Weighted vector-valued bounds for a class of multilinear singular integral operators and applications

Jiecheng Chen|Guo-En Hu
대한수학회 (The Korean Mathematical Society)|2018.05|대한수학회지 (Journal of the KMS)|제55권 제3호|pp. 671-694

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참고문헌

키워드

commutatorweighted vector-valued inequalitymultilinear singular integral operatornon-smooth kernelmultiple weight
주제분류수학

초록

In this paper, we investigate the weighted vector-valued \linebreak bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times L^{p_m}(l^{q_m};\,\mathbb{R}^n,w_m)$ to $L^{p}(l^q;\mathbb{R}^n$, $\nu_{\vec{w}})$, with $p_1,\ldots,p_m$, $q_1,\ldots,q_m\in (1,\,\infty)$, $1/p=1/p_1+\dots+1/p_m$, $1/q=1/q_1+\dots+1/q_m$ and $\vec{w}=(w_1,\ldots,w_m)$ a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators. As applications, we obtain some new weighted estimates for the Calder\'on commutator.

Abstract

In this paper, we investigate the weighted vector-valued \linebreak bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times L^{p_m}(l^{q_m};\,\mathbb{R}^n,w_m)$ to $L^{p}(l^q;\mathbb{R}^n$, $\nu_{\vec{w}})$, with $p_1,\ldots,p_m$, $q_1,\ldots,q_m\in (1,\,\infty)$, $1/p=1/p_1+\dots+1/p_m$, $1/q=1/q_1+\dots+1/q_m$ and $\vec{w}=(w_1,\ldots,w_m)$ a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators. As applications, we obtain some new weighted estimates for the Calder\'on commutator.

참고문헌 (36)

  1. S. M. Buckley, "Estimates for operator norms on weighted spaces and reverse Jensen inequalities", Trans. Amer. Math. Soc., 340(1), pp.253~272 (1993)[DOI]
  2. A. Bui, "Weighted bounds for multilinear operators with non-smooth kernels", arxiv:1506.07752, ~
  3. A.-P. Calderon, "Commutators of singular integral operators", Proc. Nat. Acad. Sci. U.S.A., 53, pp.1092~1099 (1965)[DOI]
  4. C. P. Calderon, "On commutators of singular integrals", Studia Math., 53(2), pp.139~174 (1975)[DOI]
  5. D. Chung, "Sharp bounds for general commutators on weighted Lebesgue spaces", Trans. Amer. Math. Soc., 364(3), pp.1163~1177 (2012)[DOI]
  6. R. R. Coifman, "On commutators of singular integrals and bilinear singular integrals", Trans. Amer. Math. Soc., 212, pp.315~331 (1975)[DOI]
  7. R. R. Coifman, "Au dela des operateurs pseudo-differentiels, Asterisque, 57" (1978)
  8. J. M. Conde-Alonso, "A pointwise estimate for positive dyadic shifts and some applications", Math. Ann., 365(3-4), pp.1111~1135 (2016)[DOI]
  9. D. Cruz-Uribe, "Sharp weighted estimates for classical operators", Adv. Math., 229(1), pp.408~441 (2012)[DOI]
  10. W. Damian, "New bounds for bilinear Calderon-Zygmund operators and applications", arxiv:1512.02400, ~
  11. O. Dragicevic, "Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces", Publ. Mat., 49, pp.73~91 (2005)[DOI]
  12. X. Duong, "Maximal operator for multilinear singular integrals with non-smooth kernels", Indiana Univ. Math. J., 58(6), pp.2517~2541 (2009)[DOI]
  13. X. T. Duong, "Multilinear operators with non-smooth kernels and commutators of singular integrals", Trans. Amer. Math. Soc., 362(4), pp.2089~2113 (2010)
  14. C. Fefferman, "Some maximal inequalities", Amer. J. Math., 93, pp.107~115 (1971)[DOI]
  15. L. Grafakos, "Modern Fourier Analysis" (2009), Springer
  16. L. Grafakos, "Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels", Proc. Roy. Soc. Edinburgh Sect. A, 141(4), pp.755~775 (2011)[DOI]
  17. L. Grafakos, "Multilinear Calderon-Zygmund theory", Adv. Math., 165(1), pp.124~164 (2002)[DOI]
  18. G. Hu, "A Cotlar type inequality for the multilinear singular integral operators and its applications", J. Math. Anal. Appl., 290(2), pp.639~653 (2004)[DOI]
  19. G. Hu, "Weighted vector-valued inequalities for a class of multilinear singular integral operators", Proc. Edinb. Math. Soc.
  20. G. E. Hu, "Weighted norm inequalities with general weights for the commutator of Calderon", Acta Math. Sin. (Engl. Ser.), 29(3), pp.505~514 (2013)[DOI]
  21. T. P. Hytonen, "The sharp weighted bound for general Calderon-Zygmund operators", Ann. of Math., 175(3), pp.1473~1506 (2012)[DOI]
  22. T. P. Hytonen, "Sharp weighted bounds for the q-variation of singular integrals", Bull. Lond. Math. Soc., 45(3), pp.529~540 (2013)[DOI]
  23. T. Hytonen, "Sharp weighted bounds involving <TEX>$A_{\infty}$<", TEX>, 2013, pp.6(4), 777~818
  24. A. K. Lerner, "On pointwise estimates involving sparse operators", New York J. Math., 22, pp.341~349 (2016)
  25. K. Lerner, "Intuitive dyadic calculus: the basics", arXiv:1508.05639, ~
  26. A. Lerner, "New maximal functions and multiple weights for the multilinear Calderon-Zygmund theory", Adv. Math., 220(4), pp.1222~1264 (2009)[DOI]
  27. A. K. Lerner, "On pointwise and weighted estimates for commutators of Calderon-Zygmund operators", Adv. Math., 319, pp.153~181 (2017)[DOI]
  28. K. Li, "Sparse domination theorem for mltilinear singular integral operators with <TEX>$L^r$<", TEX>-Hormander condition, 2017, pp.95, 124034~
  29. K. Li, "The sharp weighted bound for multilinear maximal functions and Calderon-Zygmund operators", J. Fourier Anal. Appl., 20(4), pp.751~765 (2014)[DOI]
  30. K. Li, "Weak and strong type weighted estimates for multilinear Calderon-Zygmund operators", Adv. Math., 254, pp.736~771 (2014)[DOI]
  31. B. Muckenhoupt, "Weighted norm inequalities for the Hardy maximal function", Trans. Amer. Math. Soc., 165, pp.207~226 (1972)[DOI]
  32. C. Perez, "End-point estimates for iterated commutators of multilinear singular integrals", Bull. Lond. Math. Soc., 46(1), pp.26~42 (2014)[DOI]
  33. S. Petermichl, "The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical <TEX>$A_p$<", TEX> characteristic, 2007, pp.129(5), 1355~1375
  34. S. Petermichl, "The sharp weighted bound for the Riesz transforms", Proc. Amer. Math. Soc., 136(4), pp.1237~1249 (2008)
  35. M. M. Rao, "Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146" (1991), Marcel Dekker, Inc.
  36. E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43" (1993), Princeton University Press

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