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Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates

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  • Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{\varphi,L}(\mathbb{R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schrödinger operators on $\mathbb{R}^n$ with non-negative potentials belonging to the reverse Hölder class, and second-order divergence form elliptic operators on $\mathbb{R}^n$ with bounded measurable real coefficients.
    Mathematics Subject Classification: Primary: 42B25; Secondary: 42B35, 46E30.

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  • [1]

    P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005.

    [2]

    P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $R^n$, J. Funct. Anal., 201 (2003), 148-184.doi: 10.1016/S0022-1236(03)00059-4.

    [3]

    P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque, 249 (1998), viii+172 pp.

    [4]

    T. A. Bui, J. Cao, L. D. Ky, D. Yang and S. Yang, Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129.

    [5]

    F. Cacciafesta and P. D'Ancona, Weighted $L^p$ estimates for powers of selfadjoint operators, Adv. Math., 229 (2012), 501-530.doi: 10.1016/j.aim.2011.09.007.

    [6]

    A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces, Adv. Math., 25 (1977), 216-225.

    [7]

    J. Cao, D.-C. Chang, D. Yang and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463.doi: 10.3934/cpaa.2014.13.1435.

    [8]

    J. Cao, S. Mayboroda and D. Yang, Local Hardy spaces associated with inhomogeneous higher order elliptic operators, Anal. Appl. (Singap.), (2015).doi: 10.1142/S0219530515500189.

    [9]

    J. Cao, S. Mayboroda and D. Yang, Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators, Forum Math. doi: 10.1515/forum-2014-0127.

    [10]

    D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc., 347 (1995), 2941-2960.doi: 10.2307/2154763.

    [11]

    X. T. Duong and J. Li, Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal., 264 (2013), 1409-1437.doi: 10.1016/j.jfa.2013.01.006.

    [12]

    J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators, in Fourier analysis and related topics, Banach Center Publ., 56, Polish Acad. Sci., Warsaw, (2002), 45-53.

    [13]

    C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.

    [14]

    L. Grafakos, Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics 250, Springer, New York, 2009.doi: 10.1007/978-0-387-09434-2.

    [15]

    S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc., 214 (2011), no. 1007, vi+78 pp.doi: 10.1090/S0065-9266-2011-00624-6.

    [16]

    S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann., 344 (2009), 37-116.doi: 10.1007/s00208-008-0295-3.

    [17]

    S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), no. 6, 1350029, 37 pp.doi: 10.1142/S0219199713500296.

    [18]

    R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal., 258 (2010), 1167-1224.doi: 10.1016/j.jfa.2009.10.018.

    [19]

    R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math., 13 (2011), 331-373.doi: 10.1142/S0219199711004221.

    [20]

    R. Jiang, Da. Yang and Do. Yang, Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators, Forum Math., 24 (2012), 471-494.doi: 10.1515/form.2011.067.

    [21]

    R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273.doi: 10.4171/RMI/50.

    [22]

    L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory, 78 (2014), 115-150.doi: 10.1007/s00020-013-2111-z.

    [23]

    L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958.doi: 10.1090/S0002-9947-2012-05727-8.

    [24]

    S. Liu and L. Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators, J. Funct. Anal., 265 (2013), 2709-2723.doi: 10.1016/j.jfa.2013.08.003.

    [25]

    J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983.doi: 10.1007/BFb0072210.

    [26]

    M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.doi: 10.1080/03601239109372748.

    [27]

    L. Song and L. Yan, Riesz transforms associated to Schrödinger operators on weighted Hardy spaces, J. Funct. Anal., 259 (2010), 1466-1490.doi: 10.1016/j.jfa.2010.05.015.

    [28]

    L. Song and L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math., 287 (2016), 463-484.doi: 10.1016/j.aim.2015.09.026.

    [29]

    E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993.

    [30]

    E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces, Acta Math., 103 (1960), 25-62.

    [31]

    L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc., 360 (2008), 4383-4408.doi: 10.1090/S0002-9947-08-04476-0.

    [32]

    Da. Yang and Do. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators, Front. Math. China, 10 (2015), 1203-1232.doi: 10.1007/s11464-015-0432-8.

    [33]

    D. Yang and S. Yang, Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of $\mathbbR^n$, Indiana Univ. Math. J., 61 (2012), 81-129.doi: 10.1512/iumj.2012.61.4535.

    [34]

    D. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators and their applications, J. Geom. Anal., 24 (2014), 495-570.doi: 10.1007/s12220-012-9344-y.

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