2014, 13(4): 1435-1463. doi: 10.3934/cpaa.2014.13.1435

Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

2. 

Department of Mathematics and Department of Computer Science, Georgetown University, Washington D.C. 20057, United States

3. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875

4. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  March 2013 Revised  January 2014 Published  February 2014

Let $L:=-\Delta+V$ be a Schrödinger operator with the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q_0}(R^n)$ for some $q_0\in[n,\infty)$ with $n\geq 3$, and $\varphi: R^n\times[0,\infty)\to[0,\infty)$ a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in A_{\infty}(R^n)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index $i(\varphi)\in (\frac{n}{n+1},1]$. In this article, the authors prove that the second order Riesz transform $\nabla^2L^{-1}$ associated with $L$ is bounded from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi,L}(R^n)$, to the Musielak-Orlicz-Hardy space $H_{\varphi}(R^n)$, via establishing an atomic characterization of $H_{\varphi,L}(R^n)$. As an application, the authors prove that the operator $VL^{-1}$ is bounded on the Musielak-Orlicz-Hardy space $H_{\varphi,L}(R^n)$, which further gives the maximal inequality associated with $L$ in $H_{\varphi,L}(R^n)$. All these results are new even when $\varphi(x,t):=t^p$, with $p\in(\frac{n}{n+1},1]$, for all $x\in R^n$ and $t\in[0,\infty)$.
Citation: 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
References:
[1]

P. Auscher and B. Ben Ali, Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1975.

[2]

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X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces,, \emph{Unpublished Manuscript}, (2005).

[3]

P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds,, \emph{J. Geom. Anal.}, 18 (2008), 192. doi: 10.1007/s12220-007-9003-x.

[4]

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

[5]

N. Badr and B. Ben Ali, $L^p$ boundedness of the Riesz transform related to Schrödinger operators on a manifold,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}, 8 (2009), 725.

[6]

A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces,, \emph{Publ. Mat.}, 54 (2010), 341. doi: 10.5565/PUBLMAT_54210_03.

[7]

A. Bonami and S. Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces,, \emph{Colloq. Math.}, 118 (2010), 107. doi: 10.4064/cm118-1-5.

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A. Bonami, S. Grellier and L. D. Ky, Paraproducts and products of functions in $BMO(\mathbbR^n)$ and $H^1(\mathbbR^n)$ through wavelets,, \emph{J. Math. Pures Appl. (9)}, 97 (2012), 230. doi: 10.1016/j.matpur.2011.06.002.

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A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in $BMO$ and $H^1$,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1405.

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J. Cao, D.-C. Chang, D. Yang and S. Yang, Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 4729. doi: 10.1090/S0002-9947-2013-05832-1.

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R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis,, \emph{J. Funct. Anal.}, 62 (1985), 304. doi: 10.1016/0022-1236(85)90007-2.

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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.

[14]

L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,, \emph{Bull. Sci. Math.}, 129 (2005), 657. doi: 10.1016/j.bulsci.2003.10.003.

[15]

L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability,, \emph{J. Funct. Anal.}, 256 (2009), 1731. doi: 10.1016/j.jfa.2009.01.017.

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X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 183. doi: 10.4171/RMI/718.

[17]

X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds,, \emph{J. Amer. Math. Soc.}, 18 (2005), 943. doi: 10.1090/S0894-0347-05-00496-0.

[18]

X. T. Duong and L. Yan, Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates,, \emph{J. Math. Soc. Japan}, 63 (2011), 295.

[19]

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

[20]

J. Dziubański and J. Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potential from reverse Hölder classes,, \emph{Colloq. Math.}, 98 (2003), 5. doi: 10.4064/cm98-1-2.

[21]

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

[22]

J. García-Cuerva, Weighted $H^p$ spaces,, \emph{Dissertationes Math. (Rozprawy Mat.)}, 162 (1979), 1.

[23]

J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics,, Amsterdam, (1985).

[24]

F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping,, \emph{Acta Math.}, 130 (1973), 265.

[25]

L. Grafakos, Modern Fourier Analysis,, 2$^{nd}$ edition, (2009). doi: 10.1007/978-0-387-09434-2.

[26]

D. Goldberg, A local version of real Hardy spaces,, \emph{Duke Math. J.}, 46 (1979), 27.

[27]

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).

[28]

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.

[29]

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.

[30]

S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces,, \emph{Ann. Sci. \'Ecole Norm. Sup. (4)}, 44 (2011), 723.

[31]

S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation,, \emph{Duke Math. J.}, 47 (1980), 959.

[32]

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.

[33]

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.

[34]

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.

[35]

R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators,, \emph{Sci. China Ser. A}, 52 (2009), 1042. doi: 10.1007/s11425-008-0136-6.

[36]

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

[37]

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.

[38]

J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math., (1034).

[39]

E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation,, \emph{J. Math. Soc. Japan}, 37 (1985), 207. doi: 10.2969/jmsj/03720207.

[40]

E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005).

[41]

M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991).

[42]

M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker, (2002). doi: 10.1201/9780203910863.

[43]

S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller,, \emph{Comm. Partial Differential Equations}, 19 (1994), 277. doi: 10.1080/03605309408821017.

[44]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential,, \emph{Ann. Inst. Fourier (Grenoble)}, 45 (1995), 513.

[45]

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.

[46]

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.

[47]

J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces,, \emph{Indiana Univ. Math. J.}, 28 (1979), 511. doi: 10.1512/iumj.1979.28.28037.

[48]

J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces,, Lecture Notes in Math., (1381).

[49]

S. Sugano, $L^p$ estimates for some Schrödinger type operators and a Calderón-Zygmund operator of Schrödinger type,, \emph{Tokyo J. Math.}, 30 (2007), 179. doi: 10.3836/tjm/1184963655.

[50]

L. Tang, Weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators, preprint,, \arXiv{1109.0100}., ().

[51]

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.

[52]

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.

[53]

D. Yang and S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of $\mathbbR^n$,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 233. doi: 10.4171/RMI/719.

[54]

D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications,, \emph{Sci. China Math.}, 55 (2012), 1677. doi: 10.1007/s11425-012-4377-z.

[55]

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.

[56]

J. Zhong, The Sobolev estimates for some Schrödinger type operators,, \emph{Math. Sci. Res. Hot-Line}, 3 (1999), 1.

show all references

References:
[1]

P. Auscher and B. Ben Ali, Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1975.

[2]

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

[3]

P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds,, \emph{J. Geom. Anal.}, 18 (2008), 192. doi: 10.1007/s12220-007-9003-x.

[4]

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

[5]

N. Badr and B. Ben Ali, $L^p$ boundedness of the Riesz transform related to Schrödinger operators on a manifold,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}, 8 (2009), 725.

[6]

A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces,, \emph{Publ. Mat.}, 54 (2010), 341. doi: 10.5565/PUBLMAT_54210_03.

[7]

A. Bonami and S. Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces,, \emph{Colloq. Math.}, 118 (2010), 107. doi: 10.4064/cm118-1-5.

[8]

A. Bonami, S. Grellier and L. D. Ky, Paraproducts and products of functions in $BMO(\mathbbR^n)$ and $H^1(\mathbbR^n)$ through wavelets,, \emph{J. Math. Pures Appl. (9)}, 97 (2012), 230. doi: 10.1016/j.matpur.2011.06.002.

[9]

A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in $BMO$ and $H^1$,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1405.

[10]

J. Cao, D.-C. Chang, D. Yang and S. Yang, Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 4729. doi: 10.1090/S0002-9947-2013-05832-1.

[11]

R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces,, \emph{J. Math. Pures Appl. (9)}, 72 (1993), 247.

[12]

R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis,, \emph{J. Funct. Anal.}, 62 (1985), 304. doi: 10.1016/0022-1236(85)90007-2.

[13]

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.

[14]

L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,, \emph{Bull. Sci. Math.}, 129 (2005), 657. doi: 10.1016/j.bulsci.2003.10.003.

[15]

L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability,, \emph{J. Funct. Anal.}, 256 (2009), 1731. doi: 10.1016/j.jfa.2009.01.017.

[16]

X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 183. doi: 10.4171/RMI/718.

[17]

X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds,, \emph{J. Amer. Math. Soc.}, 18 (2005), 943. doi: 10.1090/S0894-0347-05-00496-0.

[18]

X. T. Duong and L. Yan, Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates,, \emph{J. Math. Soc. Japan}, 63 (2011), 295.

[19]

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

[20]

J. Dziubański and J. Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potential from reverse Hölder classes,, \emph{Colloq. Math.}, 98 (2003), 5. doi: 10.4064/cm98-1-2.

[21]

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

[22]

J. García-Cuerva, Weighted $H^p$ spaces,, \emph{Dissertationes Math. (Rozprawy Mat.)}, 162 (1979), 1.

[23]

J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics,, Amsterdam, (1985).

[24]

F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping,, \emph{Acta Math.}, 130 (1973), 265.

[25]

L. Grafakos, Modern Fourier Analysis,, 2$^{nd}$ edition, (2009). doi: 10.1007/978-0-387-09434-2.

[26]

D. Goldberg, A local version of real Hardy spaces,, \emph{Duke Math. J.}, 46 (1979), 27.

[27]

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).

[28]

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.

[29]

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.

[30]

S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces,, \emph{Ann. Sci. \'Ecole Norm. Sup. (4)}, 44 (2011), 723.

[31]

S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation,, \emph{Duke Math. J.}, 47 (1980), 959.

[32]

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.

[33]

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.

[34]

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.

[35]

R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators,, \emph{Sci. China Ser. A}, 52 (2009), 1042. doi: 10.1007/s11425-008-0136-6.

[36]

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

[37]

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.

[38]

J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math., (1034).

[39]

E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation,, \emph{J. Math. Soc. Japan}, 37 (1985), 207. doi: 10.2969/jmsj/03720207.

[40]

E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005).

[41]

M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991).

[42]

M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker, (2002). doi: 10.1201/9780203910863.

[43]

S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller,, \emph{Comm. Partial Differential Equations}, 19 (1994), 277. doi: 10.1080/03605309408821017.

[44]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential,, \emph{Ann. Inst. Fourier (Grenoble)}, 45 (1995), 513.

[45]

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.

[46]

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.

[47]

J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces,, \emph{Indiana Univ. Math. J.}, 28 (1979), 511. doi: 10.1512/iumj.1979.28.28037.

[48]

J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces,, Lecture Notes in Math., (1381).

[49]

S. Sugano, $L^p$ estimates for some Schrödinger type operators and a Calderón-Zygmund operator of Schrödinger type,, \emph{Tokyo J. Math.}, 30 (2007), 179. doi: 10.3836/tjm/1184963655.

[50]

L. Tang, Weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators, preprint,, \arXiv{1109.0100}., ().

[51]

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.

[52]

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.

[53]

D. Yang and S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of $\mathbbR^n$,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 233. doi: 10.4171/RMI/719.

[54]

D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications,, \emph{Sci. China Math.}, 55 (2012), 1677. doi: 10.1007/s11425-012-4377-z.

[55]

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.

[56]

J. Zhong, The Sobolev estimates for some Schrödinger type operators,, \emph{Math. Sci. Res. Hot-Line}, 3 (1999), 1.

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