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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

Dachun Yang 1, and  Sibei Yang 2,

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.
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X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces,, \emph{Unpublished Manuscript}, (2005). 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

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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

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J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45. Google Scholar

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C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137. Google Scholar

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L. Grafakos, Modern Fourier Analysis,, Second edition, (2009). doi: 10.1007/978-0-387-09434-2. Google Scholar

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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

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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

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