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Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

Published online by Cambridge University Press:  11 January 2016

Tri Dung Tran
Tri Dung Tran*
Affiliation:
Department of Mathematics, University of Pedagogy, Ho Chi Minh city, [email protected], [email protected]
*
Department of Mathematics, Macquarie University, NSW 2109, Australia
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Abstract

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Let L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

Keywords

42B2035B6535K0542B2547B3858J35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 216 , 2014 , pp. 71 - 110
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Auscher, P., On necessary and sufficient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on ℝn and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871. MR 2292385. DOI 10.1090/memo/0871.Google Scholar
[2] Auscher, P., Duong, X. T., and McIntosh, A., Boundedness of Banach space valued singular integral operators and Hardy spaces, in preparation.Google Scholar
[3] Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., and Tchamitchian, P., The solution of the Kato square root problem for second order elliptic operators on ∝n , Ann. of Math. (2) 156 (2002), 633–654. MR 1933726. DOI 10.2307/3597201.CrossRefGoogle Scholar
[4] Auscher, P., McIntosh, A., and Russ, E., Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192–248. MR 2365673. DOI 10.1007/s12220-007-9003-x.CrossRefGoogle Scholar
[5] Auscher, P. and Russ, E., Hardy spaces and divergence operators on strongly Lipschitz domains of ℝn , J. Funct. Anal. 201 (2003), 148–184. MR 1986158. DOI 10.1016/S0022-1236(03)00059-4.CrossRefGoogle Scholar
[6] Auscher, P. and Tchamitchian, P., Square Root Problem for Divergence Operators and Related Topics, Astérisque 249, Soc. Math. France, Paris, 1998. MR 1651262.Google Scholar
[7] Badr, N., Jiménez del Toro, A., and Martell, J. M., Lp self-improvement of generalized Poincaré inequalities in spaces of homogeneous type, J. Funct. Anal. 260 (2011), 3147–3188. MR 2776565. DOI 10.1016/j.jfa.2011.01.014.CrossRefGoogle Scholar
[8] Bernicot, F., Use of Hardy spaces and interpolation, C. R. Math. Acad. Sci. Paris 346 (2008), 745–748. MR 2427074. DOI 10.1016/j.crma.2008.05.009.CrossRefGoogle Scholar
[9] Bernicot, F. and Martell, J. M., Self-improving properties for abstract Poincaré type inequalities, preprint, arXiv:1107.2260v1 [math.CA].CrossRefGoogle Scholar
[10] Bernicot, F. and Zhao, J., New abstract Hardy spaces, J. Funct. Anal. 255 (2008), 1761–1796. MR 2442082. DOI 10.1016/j.jfa.2008.06.018.CrossRefGoogle Scholar
[11] Bernicot, F. and Zhao, J., Abstract framework for John-Nirenberg inequalities and applications to Hardy spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), 475–501. MR 3059835.Google Scholar
[12] Bui, T. A., Cao, J., Ky, L. D., Yang, D., and Yang, S., Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces 1 (2012), 69–129. MR 3108869.CrossRefGoogle Scholar
[13] Coifman, R. R., Meyer, Y., and Stein, E. M., Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335. MR 0791851. DOI 10.1016/0022-1236(85)90007-2.CrossRefGoogle Scholar
[14] Diening, L., Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), 657–700. MR 2166733. DOI 10.1016/j.bulsci.2003.10.003.CrossRefGoogle Scholar
[15] Duong, X. T., Xiao, J., and Yan, L., Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007), 87–111. MR 2296729. DOI 10.1007/s00041-006-6057-2.CrossRefGoogle Scholar
[16] Duong, X. T. and Yan, L., Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), 943–973. MR 2163867. DOI 10.1090/S0894-0347-05-00496-0.CrossRefGoogle Scholar
[17] Duong, X. T. and Yan, L., New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), 1375–1420. MR 2162784. DOI 10.1002/cpa.20080.CrossRefGoogle Scholar
[18] Fefferman, C. and Stein, E. M., Hp spaces of several variables, Acta Math. 129 (1972), 137–193. MR 0447953.CrossRefGoogle Scholar
[19] Harboure, E., Salinas, O., and Viviani, B., A look at BMOϕ (ω) through Carleson measures, J. Fourier Anal. Appl. 13 (2007), 267–284. MR 2334610. DOI 10.1007/s00041-005-5044-3.CrossRefGoogle Scholar
[20] Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., and Yan, L., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007. MR 2868142. DOI 10.1090/S0065-9266-2011-00624-6.Google Scholar
[21] Hofmann, S. and Martell, J. M., Lp bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), 497–515. MR 2006497. DOI 10.5565/PUBLMAT_47203_12.CrossRefGoogle Scholar
[22] Hofmann, S. and Mayboroda, S., Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37–116. MR 2481054. DOI 10.1007/s00208-008-0295-3.CrossRefGoogle Scholar
[23] Janson, S., Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959–982. MR 0596123.CrossRefGoogle Scholar
[24] Jiang, R. and Yang, D., New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 1167–1224. MR 2565837. DOI 10.1016/j.jfa.2009.10.018.CrossRefGoogle Scholar
[25] Jiang, R. and Yang, D., Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math. 13 (2011), 331–373. MR 2794490. DOI 10.1142/S0219199711004221.CrossRefGoogle Scholar
[26] Jiang, R., Yang, D., and Zhou, Y., Orlicz-Hardy spaces associated with operators, Sci. China Ser. A 52 (2009), 1042–1080. MR 2505009. DOI 10.1007/s11425-008-0136-6.CrossRefGoogle Scholar
[27] Jiménez del Toro, A. and Martell, J. M., Self-improvement of Poincaré type inequalities associated with approximations of the identity and semigroups, Potential Anal. 38 (2013), 805–841. MR 3034601. DOI 10.1007/s11118-012-9298-5.CrossRefGoogle Scholar
[28] Ky, L. D., New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), 115–150. MR 3147406. DOI 10.1007/s00020-013-2111-z.CrossRefGoogle Scholar
[29] Liang, Y., Yang, D., and Yang, S., Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates, Sci. China Math. 54 (2011), 2395–2426. MR 2859702. DOI 10.1007/s11425-011-4294-6.CrossRefGoogle Scholar
[30] Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983. MR 0724434.Google Scholar
[31] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. MR 1232192.Google Scholar
[32] Stein, E. M. and Weiss, G., On the theory of harmonic functions of several variables, I: The theory of Hp-spaces, Acta Math. 103 (1960), 25–62. MR 0121579.CrossRefGoogle Scholar
[33] Strömberg, J.-O., Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511–544. MR 0529683. DOI 10.1512/iumj.1979.28.28037.CrossRefGoogle Scholar
[34] Viviani, B. E., An atomic decomposition of the predual of BMO(ρ), Rev. Mat. Iberoam. 3 (1987), 401–425. MR 0996824. DOI 10.4171/RMI/56.CrossRefGoogle Scholar
[35] Yan, L., Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4383–4408. MR 2395177. DOI 10.1090/S0002-9947-08-04476-0.CrossRefGoogle Scholar
[36] Yang, D. and Yang, S., Musielak-Orlicz-Hardy spaces associated with operators and their applications, J. Geom. Anal. 24 (2014), 495–570. MR 3145932. DOI 10.1007/s12220012-9344-y.CrossRefGoogle Scholar
[37] Yosida, K., Functional Analysis, 6th ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1980. MR 0617913.Google Scholar