Self-improving properties for abstract Poincaré type inequalities

Bernicot, Frédéric; Martell, José María

doi:10.1090/S0002-9947-2014-06315-0

Self-improving properties for abstract Poincaré type inequalities
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by Frédéric Bernicot and José María Martell PDF

Trans. Amer. Math. Soc. 367 (2015), 4793-4835 Request permission

Abstract:

We study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the identity, semigroups or mean value operators) that have off-diagonal decay in some range. Our results provide a unified theory that is applicable to the classical Poincaré inequalities, and furthermore it includes oscillations defined in terms of semigroups associated with second order elliptic operators as those in the Kato conjecture. In this latter situation we obtain a direct proof of the John-Nirenberg inequality for the associated $BMO$ and Lipschitz spaces of S. Hofmann, S. Mayboroda, and A. McIntosh.

References

Pascal Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\Bbb R^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385, DOI 10.1090/memo/0871
Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), no. 2, 265–316. MR 2316480, DOI 10.1007/s00028-007-0288-9
Nadine Badr, Ana Jiménez-del-Toro, and José María Martell, $L^p$ self-improvement of generalized Poincaré inequalities in spaces of homogeneous type, J. Funct. Anal. 260 (2011), no. 11, 3147–3188. MR 2776565, DOI 10.1016/j.jfa.2011.01.014
Frédéric Bernicot and Jiman Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008), no. 7, 1761–1796. MR 2442082, DOI 10.1016/j.jfa.2008.06.018
Frédéric Bernicot and Jiman Zhao, On maximal $L^p$-regularity, J. Funct. Anal. 256 (2009), no. 8, 2561–2586. MR 2502526, DOI 10.1016/j.jfa.2009.01.018
Frédéric Bernicot and Jiman Zhao, Abstract framework for John-Nirenberg inequalities and applications to Hardy spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 3, 475–501. MR 3059835
D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
Donggao Deng, Xuan Thinh Duong, and Lixin Yan, A characterization of the Morrey-Campanato spaces, Math. Z. 250 (2005), no. 3, 641–655. MR 2179615, DOI 10.1007/s00209-005-0769-x
Xuan Thinh Duong and Lixin Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), no. 10, 1375–1420. MR 2162784, DOI 10.1002/cpa.20080
Xuan Thinh Duong and Lixin Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973. MR 2163867, DOI 10.1090/S0894-0347-05-00496-0
Bruno Franchi, Carlos Pérez, and Richard L. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Funct. Anal. 153 (1998), no. 1, 108–146. MR 1609261, DOI 10.1006/jfan.1997.3175
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
Juha Heinonen and Pekka Koskela, From local to global in quasiconformal structures, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), no. 2, 554–556. MR 1372507, DOI 10.1073/pnas.93.2.554
Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
Steve Hofmann and José María Martell, $L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), no. 2, 497–515. MR 2006497, DOI 10.5565/PUBLMAT_{4}7203_{1}2
Steve Hofmann and Svitlana Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37–116. MR 2481054, DOI 10.1007/s00208-008-0295-3
Steve Hofmann, Svitlana Mayboroda, and Alan McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 5, 723–800 (English, with English and French summaries). MR 2931518, DOI 10.24033/asens.2154
Ana Jiménez-del-Toro, Exponential self-improvement of generalized Poincaré inequalities associated with approximations of the identity and semigroups, Trans. Amer. Math. Soc. 364 (2012), no. 2, 637–660. MR 2846346, DOI 10.1090/S0002-9947-2011-05344-4
Ana Jiménez-del-Toro and José María Martell, Self-improvement of Poincaré type inequalities associated with approximations of the identity and semigroups, Potential Anal. 38 (2013), no. 3, 805–841. MR 3034601, DOI 10.1007/s11118-012-9298-5
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
Paul MacManus and Carlos Pérez, Trudinger inequalities without derivatives, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1997–2012. MR 1881027, DOI 10.1090/S0002-9947-02-02918-5
José María Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004), no. 2, 113–145. MR 2033231, DOI 10.4064/sm161-2-2