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Sharp weighted bounds for one-sided operators

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Published/Copyright: May 17, 2017
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Abstract

In this paper, we establish sharp weighted bounds (Buckley-type theorems) for one-sided maximal and fractional integral operators in terms of one-sided Ap characteristics.

MSC 2010: 42B20; 42B25

Acknowledgements

A part of this work was carried out at Riphah International University, Islamabad. The third-named author thanks the Vice Chancellor of Riphah International University, Islamabad, Pakistan.

References

[1] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the Riemann–Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), no. 2, 547–558. 10.1090/S0002-9947-1988-0930071-4Search in Google Scholar

[2] K. Astala, T. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27–56. 10.1215/S0012-7094-01-10713-8Search in Google Scholar

[3] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272. 10.1090/S0002-9947-1993-1124164-0Search in Google Scholar

[4] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. 10.4064/sm-51-3-241-250Search in Google Scholar

[5] D. Cruz-Uribe and K. Moen, A fractional Muckenhoupt–Wheeden theorem and its consequences, Integral Equations Operator Theory 76 (2013), no. 3, 421–446. 10.1007/s00020-013-2059-zSearch in Google Scholar

[6] O. Dragičević, L. Grafakos, M. C. Pereyra and S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), no. 1, 73–91. 10.5565/PUBLMAT_49105_03Search in Google Scholar

[7] D. E. Edmunds, V. Kokilashvili and A. Meskhi, Bounded and Compact Integral Operators, Math. Appl. 543, Kluwer Academic, Dordrecht, 2002. 10.1007/978-94-015-9922-1Search in Google Scholar

[8] D. E. Edmunds, V. Kokilashvili and A. Meskhi, On Fourier multipliers in weighted Triebel–Lizorkin spaces, J. Inequal. Appl. 7 (2002), no. 4, 555–591. 10.1155/S1025583402000279Search in Google Scholar

[9] R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. 10.1090/S0002-9947-1973-0312139-8Search in Google Scholar

[10] T. P. Hytönen, The sharp weighted bound for general Calderón–Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. 10.4007/annals.2012.175.3.9Search in Google Scholar

[11] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, River Edge, 1991. 10.1142/1367Search in Google Scholar

[12] V. Kokilashvili, A. Meskhi and M. A. Zaighum, On sharp weighted bounds for one-sided operators norms, Proc. A. Razmadze Math. Inst. 164 (2014), 121–129. Search in Google Scholar

[13] M. T. Lacey, K. Moen, C. Pérez and R. H. Torres, Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010), no. 5, 1073–1097. 10.1016/j.jfa.2010.02.004Search in Google Scholar

[14] A. K. Lerner, An elementary approach to several results on the Hardy–Littlewood maximal operator, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2829–2833. 10.1090/S0002-9939-08-09318-0Search in Google Scholar

[15] M. Lorente and A. de la Torre, Weighted inequalities for some one-sided operators, Proc. Amer. Math. Soc. 124 (1996), no. 3, 839–848. 10.1090/S0002-9939-96-03089-4Search in Google Scholar

[16] F. Martin-Reyes, New proofs of weighted inequalities for the one-sided Hardy–Littlewood maximal functions, Proc. Amer. Math. Soc. 117 (1993), no. 3, 691–698. 10.1090/S0002-9939-1993-1111435-2Search in Google Scholar

[17] F. J. Martin-Reyes and A. de la Torre, Two weight norm inequalities for fractional one-sided maximal operators, Proc. Amer. Math. Soc. 117 (1993), no. 2, 483–489. 10.1090/S0002-9939-1993-1110548-9Search in Google Scholar

[18] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. 10.1090/S0002-9947-1972-0293384-6Search in Google Scholar

[19] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. 10.1090/S0002-9947-1974-0340523-6Search in Google Scholar

[20] S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical Ap characteristic, Amer. J. Math. 129 (2007), no. 5, 1355–1375. 10.1353/ajm.2007.0036Search in Google Scholar

[21] S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1237–1249. 10.1090/S0002-9939-07-08934-4Search in Google Scholar

[22] S. Petermichl and A. Volberg, Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. 10.1215/S0012-9074-02-11223-XSearch in Google Scholar

[23] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), no. 1, 53–61. 10.1090/S0002-9947-1986-0849466-0Search in Google Scholar

[24] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), no. 3, 257–272. 10.4064/sm-107-3-257-272Search in Google Scholar

Received: 2015-3-23
Revised: 2016-7-1
Accepted: 2016-7-5
Published Online: 2017-5-17
Published in Print: 2017-6-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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