An abstract theory of singular operators
HTML articles powered by AMS MathViewer
- by Grigori A. Karagulyan PDF
- Trans. Amer. Math. Soc. 372 (2019), 4761-4803 Request permission
Abstract:
We introduce a class of operators on abstract measure spaces that unifies the Calderón–Zygmund operators on spaces of homogeneous type, the maximal functions, the martingale transforms, and Carleson operators. We prove that such operators can be dominated by simple sparse operators with a definite form of the domination constant. Applying these estimates, we improve on several results obtained by different authors in recent years.References
- T. Anderson and A. Vagharshakyan, A simple proof of the sharp weighted estimate for Calderń-Zygmund operators on homogeneous spaces, J. Geom. Anal. 24 (2014), no. 3, 1276–1297.
- A. Bonami and D. Lépingle, Fonction maximale et variation quadratique des martingales en présence d’un poids, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 294–306 (French).
- Stephen M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272. MR 1124164, DOI 10.1090/S0002-9947-1993-1124164-0
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- José M. Conde-Alonso and Guillermo Rey, A pointwise estimate for positive dyadic shifts and some applications, Math. Ann. 365 (2016), no. 3-4, 1111–1135. MR 3521084, DOI 10.1007/s00208-015-1320-y
- Francesco Di Plinio and Andrei K. Lerner, On weighted norm inequalities for the Carleson and Walsh-Carleson operator, J. Lond. Math. Soc. (2) 90 (2014), no. 3, 654–674. MR 3291794, DOI 10.1112/jlms/jdu049
- Oliver Dragičević, Loukas Grafakos, María Cristina Pereyra, and Stefanie Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), no. 1, 73–91. MR 2140200, DOI 10.5565/PUBLMAT_{4}9105_{0}3
- Loukas Grafakos, José María Martell, and Fernando Soria, Weighted norm inequalities for maximally modulated singular integral operators, Math. Ann. 331 (2005), no. 2, 359–394. MR 2115460, DOI 10.1007/s00208-004-0586-2
- Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
- Tuomas Hytönen and Anna Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126 (2012), no. 1, 1–33. MR 2901199, DOI 10.4064/cm126-1-1
- Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. MR 2912709, DOI 10.4007/annals.2012.175.3.9
- Tuomas P. Hytönen and Michael T. Lacey, The $A_p$-$A_\infty$ inequality for general Calderón-Zygmund operators, Indiana Univ. Math. J. 61 (2012), no. 6, 2041–2092. MR 3129101, DOI 10.1512/iumj.2012.61.4777
- Tuomas P. Hytönen, Michael T. Lacey, Henri Martikainen, Tuomas Orponen, Maria Carmen Reguera, Eric T. Sawyer, and Ignacio Uriarte-Tuero, Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on $A_p$ weighted spaces, J. Anal. Math. 118 (2012), no. 1, 177–220. MR 2993026, DOI 10.1007/s11854-012-0033-3
- Tuomas P. Hytönen, Michael T. Lacey, and Carlos Pérez, Sharp weighted bounds for the $q$-variation of singular integrals, Bull. Lond. Math. Soc. 45 (2013), no. 3, 529–540. MR 3065022, DOI 10.1112/blms/bds114
- Tuomas Hytönen and Carlos Pérez, Sharp weighted bounds involving $A_\infty$, Anal. PDE 6 (2013), no. 4, 777–818. MR 3092729, DOI 10.2140/apde.2013.6.777
- Tuomas Hytönen, Carlos Pérez, and Ezequiel Rela, Sharp reverse Hölder property for $A_\infty$ weights on spaces of homogeneous type, J. Funct. Anal. 263 (2012), no. 12, 3883–3899. MR 2990061, DOI 10.1016/j.jfa.2012.09.013
- Tuomas P. Hytönen, Luz Roncal, and Olli Tapiola, Quantitative weighted estimates for rough homogeneous singular integrals, Israel J. Math. 218 (2017), no. 1, 133–164. MR 3625128, DOI 10.1007/s11856-017-1462-6
- M. Izumisawa and N. Kazamaki, Weighted norm inequalities for martingales, Tôhoku Math. J. 29 (1977), no. 1, 115–124.
- G. A. Karagulyan, Exponential estimates for the Calderón-Zygmund operator and related problems of Fourier series, Mat. Zametki 71 (2002), no. 3, 398–411 (Russian, with Russian summary); English transl., Math. Notes 71 (2002), no. 3-4, 362–373. MR 1913610, DOI 10.1023/A:1014850924850
- G. A. Karagulyan, Exponential estimates for partial sums of Fourier series in the Walsh system and the rearranged Haar system, Izv. Nats. Akad. Nauk Armenii Mat. 36 (2001), no. 5, 23–34 (2002) (Russian, with English and Russian summaries); English transl., J. Contemp. Math. Anal. 36 (2001), no. 5, 19–30 (2002). MR 1964580
- M. L. Lacey, An elementary proof of the $A_2$ bound, Israel J. Math. 217 (2017), no. 1, 181–195.
- Andrei K. Lerner, A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. Lond. Math. Soc. 42 (2010), no. 5, 843–856. MR 2721744, DOI 10.1112/blms/bdq042
- Andrei K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161. MR 3127380, DOI 10.1007/s11854-013-0030-1
- Andrei K. Lerner, A simple proof of the $A_2$ conjecture, Int. Math. Res. Not. IMRN 14 (2013), 3159–3170. MR 3085756, DOI 10.1093/imrn/rns145
- Andrei K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 341–349. MR 3484688
- A. K. Lerner and F. Nazarov, Intuitive dyadic calculus: The basics, http://arxiv.org/abs/1512.07247 (2015). Expositiones Mathematicae (to appear).
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- Kabe Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009), no. 2, 213–238. MR 2514845, DOI 10.1007/BF03191210
- Kabe Moen, Sharp weighted bounds without testing or extrapolation, Arch. Math. (Basel) 99 (2012), no. 5, 457–466. MR 3000426, DOI 10.1007/s00013-012-0453-4
- C. Pérez, S. Treil, and A. Volberg, On $A_2$ conjecture and corona decomposition of weights, http://arxiv.org/abs/1006.2630 (2010).
- S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical $A_p$ characteristic, Amer. J. Math. 129 (2007), no. 5, 1355–1375. MR 2354322, DOI 10.1353/ajm.2007.0036
- Stefanie Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1237–1249. MR 2367098, DOI 10.1090/S0002-9939-07-08934-4
- Stefanie Petermichl and Alexander Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. MR 1894362, DOI 10.1215/S0012-9074-02-11223-X
- Christoph Thiele, Sergei Treil, and Alexander Volberg, Weighted martingale multipliers in the non-homogeneous setting and outer measure spaces, Adv. Math. 285 (2015), 1155–1188. MR 3406523, DOI 10.1016/j.aim.2015.08.019
- Armen Vagharshakyan, Recovering singular integrals from Haar shifts, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4303–4309. MR 2680056, DOI 10.1090/S0002-9939-2010-10426-4
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- Grigori A. Karagulyan
- Affiliation: Faculty of Mathematics and Mechanics, Yerevan State University, 1 Alex Manoogian Street, 0025 Yerevan, Armenia
- MR Author ID: 234243
- ORCID: 0000-0001-5448-4512
- Email: g.karagulyan@ysu.am
- Received by editor(s): August 4, 2017
- Received by editor(s) in revised form: April 21, 2018, and September 27, 2018
- Published electronically: February 25, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4761-4803
- MSC (2010): Primary 42B20, 42B25; Secondary 43A85
- DOI: https://doi.org/10.1090/tran/7722
- MathSciNet review: 4009440