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

Previous Article
Some properties of positive solutions for an integral system with the double weighted Riesz potentials
CPAA Home
This Issue
Next Article
Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type

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

Abstract
Related Papers

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.

Keywords: non-negative self-adjoint operator, non-tangential maximal function, atom, Musielak-Orlicz-Hardy space, Gaussian upper bound estimate..

Mathematics Subject Classification: Primary: 42B25; Secondary: 42B35, 46E3.

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).
[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.
[3]	P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics,, \emph{Ast\'erisque}, 249 (1998).
[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.
[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.
[6]	A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces,, \emph{Adv. Math.}, 25 (1977), 216.
[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.
[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.
[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.
[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.
[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.
[12]	J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45.
[13]	C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137.
[14]	L. Grafakos, Modern Fourier Analysis,, Second edition, (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,, \emph{Mem. Amer. Math. Soc.}, 214 (2011). doi: 10.1090/S0065-9266-2011-00624-6.
[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.
[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.
[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.
[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.
[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.
[21]	R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$,, \emph{Rev. Mat. Iberoam.}, 3 (1987), 249. doi: 10.4171/RMI/50.
[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.
[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.
[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.
[25]	J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math. 1034, (1034). doi: 10.1007/BFb0072210.
[26]	M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991). doi: 10.1080/03601239109372748.
[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.
[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.
[29]	E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals,, Princeton Univ. Press, (1993).
[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.
[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.
[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.
[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.
[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.

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).
[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.
[3]	P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics,, \emph{Ast\'erisque}, 249 (1998).
[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.
[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.
[6]	A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces,, \emph{Adv. Math.}, 25 (1977), 216.
[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.
[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.
[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.
[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.
[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.
[12]	J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45.
[13]	C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137.
[14]	L. Grafakos, Modern Fourier Analysis,, Second edition, (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,, \emph{Mem. Amer. Math. Soc.}, 214 (2011). doi: 10.1090/S0065-9266-2011-00624-6.
[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.
[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.
[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.
[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.
[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.
[21]	R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$,, \emph{Rev. Mat. Iberoam.}, 3 (1987), 249. doi: 10.4171/RMI/50.
[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.
[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.
[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.
[25]	J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math. 1034, (1034). doi: 10.1007/BFb0072210.
[26]	M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991). doi: 10.1080/03601239109372748.
[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.
[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.
[29]	E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals,, Princeton Univ. Press, (1993).
[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.
[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.
[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.
[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.
[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.

[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]	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
[19]	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
[20]	Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences