# American Institute of Mathematical Sciences

November  2016, 15(6): 2135-2160. doi: 10.3934/cpaa.2016031

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

 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received  December 2015 Revised  May 2016 Published  September 2016

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.
Citation: Dachun Yang, Sibei Yang. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2135-2160. doi: 10.3934/cpaa.2016031
##### References:
 [1] P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces,, \emph{Unpublished Manuscript}, (2005). Google Scholar [2] P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $R^n$,, \emph{J. Funct. Anal.}, 201 (2003), 148. doi: 10.1016/S0022-1236(03)00059-4. Google Scholar [3] P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics,, \emph{Ast\'erisque}, 249 (1998). Google Scholar [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,, \emph{Anal. Geom. Metr. Spaces}, 1 (2013), 69. Google Scholar [5] F. Cacciafesta and P. D'Ancona, Weighted $L^p$ estimates for powers of selfadjoint operators,, \emph{Adv. Math.}, 229 (2012), 501. doi: 10.1016/j.aim.2011.09.007. Google Scholar [6] A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces,, \emph{Adv. Math.}, 25 (1977), 216. Google Scholar [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,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 1435. doi: 10.3934/cpaa.2014.13.1435. Google Scholar [8] J. Cao, S. Mayboroda and D. Yang, Local Hardy spaces associated with inhomogeneous higher order elliptic operators,, \emph{Anal. Appl. (Singap.)}, (2015). doi: 10.1142/S0219530515500189. Google Scholar [9] J. Cao, S. Mayboroda and D. Yang, Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators,, \emph{Forum Math.}, (). doi: 10.1515/forum-2014-0127. Google Scholar [10] D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes,, \emph{Trans. Amer. Math. Soc.}, 347 (1995), 2941. doi: 10.2307/2154763. Google Scholar [11] X. T. Duong and J. Li, Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus,, \emph{J. Funct. Anal.}, 264 (2013), 1409. doi: 10.1016/j.jfa.2013.01.006. Google Scholar [12] J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45. Google Scholar [13] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137. Google Scholar [14] L. Grafakos, Modern Fourier Analysis,, Second edition, (2009). doi: 10.1007/978-0-387-09434-2. Google Scholar [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,, \emph{Mem. Amer. Math. Soc.}, 214 (2011). doi: 10.1090/S0065-9266-2011-00624-6. Google Scholar [16] S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators,, \emph{Math. Ann.}, 344 (2009), 37. doi: 10.1007/s00208-008-0295-3. Google Scholar [17] S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications,, \emph{Commun. Contemp. Math.}, 15 (2013). doi: 10.1142/S0219199713500296. Google Scholar [18] R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators,, \emph{J. Funct. Anal.}, 258 (2010), 1167. doi: 10.1016/j.jfa.2009.10.018. Google Scholar [19] R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates,, \emph{Commun. Contemp. Math.}, 13 (2011), 331. doi: 10.1142/S0219199711004221. Google Scholar [20] R. Jiang, Da. Yang and Do. Yang, Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators,, \emph{Forum Math.}, 24 (2012), 471. doi: 10.1515/form.2011.067. Google Scholar [21] R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$,, \emph{Rev. Mat. Iberoam.}, 3 (1987), 249. doi: 10.4171/RMI/50. Google Scholar [22] L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators,, \emph{Integral Equations Operator Theory}, 78 (2014), 115. doi: 10.1007/s00020-013-2111-z. Google Scholar [23] L. D. Ky, Bilinear decompositions and commutators of singular integral operators,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 2931. doi: 10.1090/S0002-9947-2012-05727-8. Google Scholar [24] S. Liu and L. Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators,, \emph{J. Funct. Anal.}, 265 (2013), 2709. doi: 10.1016/j.jfa.2013.08.003. Google Scholar [25] J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math. 1034, (1034). doi: 10.1007/BFb0072210. Google Scholar [26] M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991). doi: 10.1080/03601239109372748. Google Scholar [27] L. Song and L. Yan, Riesz transforms associated to Schrödinger operators on weighted Hardy spaces,, \emph{J. Funct. Anal.}, 259 (2010), 1466. doi: 10.1016/j.jfa.2010.05.015. Google Scholar [28] L. Song and L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates,, \emph{Adv. Math.}, 287 (2016), 463. doi: 10.1016/j.aim.2015.09.026. Google Scholar [29] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals,, Princeton Univ. Press, (1993). Google Scholar [30] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces,, \emph{Acta Math.}, 103 (1960), 25. Google Scholar [31] L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 4383. doi: 10.1090/S0002-9947-08-04476-0. Google Scholar [32] Da. Yang and Do. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators,, \emph{Front. Math. China}, 10 (2015), 1203. doi: 10.1007/s11464-015-0432-8. Google Scholar [33] D. Yang and S. Yang, Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of $\mathbbR^n$,, \emph{Indiana Univ. Math. J.}, 61 (2012), 81. doi: 10.1512/iumj.2012.61.4535. Google Scholar [34] D. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators and their applications,, \emph{J. Geom. Anal.}, 24 (2014), 495. doi: 10.1007/s12220-012-9344-y. Google Scholar

show all references

##### References:
 [1] P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces,, \emph{Unpublished Manuscript}, (2005). Google Scholar [2] P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $R^n$,, \emph{J. Funct. Anal.}, 201 (2003), 148. doi: 10.1016/S0022-1236(03)00059-4. Google Scholar [3] P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics,, \emph{Ast\'erisque}, 249 (1998). Google Scholar [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,, \emph{Anal. Geom. Metr. Spaces}, 1 (2013), 69. Google Scholar [5] F. Cacciafesta and P. D'Ancona, Weighted $L^p$ estimates for powers of selfadjoint operators,, \emph{Adv. Math.}, 229 (2012), 501. doi: 10.1016/j.aim.2011.09.007. Google Scholar [6] A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces,, \emph{Adv. Math.}, 25 (1977), 216. Google Scholar [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,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 1435. doi: 10.3934/cpaa.2014.13.1435. Google Scholar [8] J. Cao, S. Mayboroda and D. Yang, Local Hardy spaces associated with inhomogeneous higher order elliptic operators,, \emph{Anal. Appl. (Singap.)}, (2015). doi: 10.1142/S0219530515500189. Google Scholar [9] J. Cao, S. Mayboroda and D. Yang, Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators,, \emph{Forum Math.}, (). doi: 10.1515/forum-2014-0127. Google Scholar [10] D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes,, \emph{Trans. Amer. Math. Soc.}, 347 (1995), 2941. doi: 10.2307/2154763. Google Scholar [11] X. T. Duong and J. Li, Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus,, \emph{J. Funct. Anal.}, 264 (2013), 1409. doi: 10.1016/j.jfa.2013.01.006. Google Scholar [12] J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45. Google Scholar [13] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137. Google Scholar [14] L. Grafakos, Modern Fourier Analysis,, Second edition, (2009). doi: 10.1007/978-0-387-09434-2. Google Scholar [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,, \emph{Mem. Amer. Math. Soc.}, 214 (2011). doi: 10.1090/S0065-9266-2011-00624-6. Google Scholar [16] S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators,, \emph{Math. Ann.}, 344 (2009), 37. doi: 10.1007/s00208-008-0295-3. Google Scholar [17] S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications,, \emph{Commun. Contemp. Math.}, 15 (2013). doi: 10.1142/S0219199713500296. Google Scholar [18] R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators,, \emph{J. Funct. Anal.}, 258 (2010), 1167. doi: 10.1016/j.jfa.2009.10.018. Google Scholar [19] R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates,, \emph{Commun. Contemp. Math.}, 13 (2011), 331. doi: 10.1142/S0219199711004221. Google Scholar [20] R. Jiang, Da. Yang and Do. Yang, Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators,, \emph{Forum Math.}, 24 (2012), 471. doi: 10.1515/form.2011.067. Google Scholar [21] R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$,, \emph{Rev. Mat. Iberoam.}, 3 (1987), 249. doi: 10.4171/RMI/50. Google Scholar [22] L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators,, \emph{Integral Equations Operator Theory}, 78 (2014), 115. doi: 10.1007/s00020-013-2111-z. Google Scholar [23] L. D. Ky, Bilinear decompositions and commutators of singular integral operators,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 2931. doi: 10.1090/S0002-9947-2012-05727-8. Google Scholar [24] S. Liu and L. Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators,, \emph{J. Funct. Anal.}, 265 (2013), 2709. doi: 10.1016/j.jfa.2013.08.003. Google Scholar [25] J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math. 1034, (1034). doi: 10.1007/BFb0072210. Google Scholar [26] M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991). doi: 10.1080/03601239109372748. Google Scholar [27] L. Song and L. Yan, Riesz transforms associated to Schrödinger operators on weighted Hardy spaces,, \emph{J. Funct. Anal.}, 259 (2010), 1466. doi: 10.1016/j.jfa.2010.05.015. Google Scholar [28] L. Song and L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates,, \emph{Adv. Math.}, 287 (2016), 463. doi: 10.1016/j.aim.2015.09.026. Google Scholar [29] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals,, Princeton Univ. Press, (1993). Google Scholar [30] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces,, \emph{Acta Math.}, 103 (1960), 25. Google Scholar [31] L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 4383. doi: 10.1090/S0002-9947-08-04476-0. Google Scholar [32] Da. Yang and Do. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators,, \emph{Front. Math. China}, 10 (2015), 1203. doi: 10.1007/s11464-015-0432-8. Google Scholar [33] D. Yang and S. Yang, Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of $\mathbbR^n$,, \emph{Indiana Univ. Math. J.}, 61 (2012), 81. doi: 10.1512/iumj.2012.61.4535. Google Scholar [34] D. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators and their applications,, \emph{J. Geom. Anal.}, 24 (2014), 495. doi: 10.1007/s12220-012-9344-y. Google Scholar
 [1] Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435 [2] Wen Deng. Resolvent estimates for a two-dimensional non-self-adjoint operator. Communications on Pure & Applied Analysis, 2013, 12 (1) : 547-596. doi: 10.3934/cpaa.2013.12.547 [3] Emmanuele DiBenedetto, Ugo Gianazza, Naian Liao. On the local behavior of non-negative solutions to a logarithmically singular equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1841-1858. doi: 10.3934/dcdsb.2012.17.1841 [4] Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353 [5] Ruirui Sun, Jinxia Li, Baode Li. Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2377-2395. doi: 10.3934/cpaa.2019107 [6] Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687 [7] Abdallah El Hamidi, Aziz Hamdouni, Marwan Saleh. On eigenelements sensitivity for compact self-adjoint operators and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 445-455. doi: 10.3934/dcdss.2016006 [8] Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481 [9] Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35 [10] Yael Ben-Haim, Simon Litsyn. A new upper bound on the rate of non-binary codes. Advances in Mathematics of Communications, 2007, 1 (1) : 83-92. doi: 10.3934/amc.2007.1.83 [11] Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831 [12] Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539 [13] Humberto Ramos Quoirin, Kenichiro Umezu. A loop type component in the non-negative solutions set of an indefinite elliptic problem. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1255-1269. doi: 10.3934/cpaa.2018060 [14] David Bourne, Howard Elman, John E. Osborn. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence. Communications on Pure & Applied Analysis, 2009, 8 (1) : 143-160. doi: 10.3934/cpaa.2009.8.143 [15] Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010 [16] Stuart S. Antman, David Bourne. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 123-142. doi: 10.3934/cpaa.2009.8.123 [17] Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61 [18] Seung Jun Chang, Jae Gil Choi. Generalized transforms and generalized convolution products associated with Gaussian paths on function space. Communications on Pure & Applied Analysis, 2020, 19 (1) : 371-389. doi: 10.3934/cpaa.2020019 [19] Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565 [20] Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018

2018 Impact Factor: 0.925