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

Jun Cao; Der-Chen Chang; Dachun Yang; Sibei Yang

doi:10.3934/cpaa.2014.13.1435

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July 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

Jun Cao ^1, , Der-Chen Chang ^2, , Dachun Yang ^3, and Sibei Yang ^4,

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

Abstract
Related Papers

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 $t\in[0,\infty)$.

Keywords: Musielak-Orlicz-Hardy space, atom, Lusin area function, Schrödinger operator, second order Riesz transform..

Mathematics Subject Classification: Primary 42B20; Secondary 42B30, 42B35, 42B25, 35J10, 42B37, 46E3.

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, Ann. Inst. Fourier (Grenoble), 57 (2007), 1975-2013. Google Scholar
[2]	P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005. Google Scholar
[3]	P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248. doi: 10.1007/s12220-007-9003-x. Google Scholar
[4]	P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $\mathbbR^n$, J. Funct. Anal., 201 (2003), 148-184. doi: 10.1016/S0022-1236(03)00059-4. Google Scholar
[5]	N. Badr and B. Ben Ali, $L^p$ boundedness of the Riesz transform related to Schrödinger operators on a manifold, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8 (2009), 725-765. Google Scholar
[6]	A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358. doi: 10.5565/PUBLMAT_54210_03. Google Scholar
[7]	A. Bonami and S. Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces, Colloq. Math., 118 (2010), 107-132. doi: 10.4064/cm118-1-5. Google Scholar
[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, J. Math. Pures Appl. (9), 97 (2012), 230-241. doi: 10.1016/j.matpur.2011.06.002. Google Scholar
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[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, Trans. Amer. Math. Soc., 365 (2013), 4729-4809. doi: 10.1090/S0002-9947-2013-05832-1. Google Scholar
[11]	R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), 72 (1993), 247-286. Google Scholar
[12]	R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304-335. doi: 10.1016/0022-1236(85)90007-2. Google Scholar
[13]	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. Google Scholar
[14]	L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657-700. doi: 10.1016/j.bulsci.2003.10.003. Google Scholar
[15]	L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal., 256 (2009), 1731-1768. doi: 10.1016/j.jfa.2009.01.017. Google Scholar
[16]	X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems, Rev. Mat. Iberoam., 29 (2013), 183-236. doi: 10.4171/RMI/718. Google Scholar
[17]	X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973. doi: 10.1090/S0894-0347-05-00496-0. Google Scholar
[18]	X. T. Duong and L. Yan, Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, J. Math. Soc. Japan, 63 (2011), 295-319. Google Scholar
[19]	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. Google Scholar
[20]	J. Dziubański and J. Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potential from reverse Hölder classes, Colloq. Math., 98 (2003), 5-38. doi: 10.4064/cm98-1-2. Google Scholar
[21]	C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. Google Scholar
[22]	J. García-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. (Rozprawy Mat.), 162 (1979), 1-63. Google Scholar
[23]	J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985. Google Scholar
[24]	F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. Google Scholar
[25]	L. Grafakos, Modern Fourier Analysis, 2nd edition, Graduate Texts in Mathematics 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2. Google Scholar
[26]	D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27-42. Google Scholar
[27]	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. Google Scholar
[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, Mem. Amer. Math. Soc., 214 (2011), no. 1007, vi+78 pp. doi: 10.1090/S0065-9266-2011-00624-6. Google Scholar
[29]	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. Google Scholar
[30]	S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4), 44 (2011), 723-800. Google Scholar
[31]	S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982. Google Scholar
[32]	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. Google Scholar
[33]	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. Google Scholar
[34]	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. Google Scholar
[35]	R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators, Sci. China Ser. A, 52 (2009), 1042-1080. doi: 10.1007/s11425-008-0136-6. Google Scholar
[36]	R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273. doi: 10.4171/RMI/50. Google Scholar
[37]	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. Google Scholar
[38]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983. Google Scholar
[39]	E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan, 37 (1985), 207-218. doi: 10.2969/jmsj/03720207. Google Scholar
[40]	E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005. Google Scholar
[41]	M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. Google Scholar
[42]	M. Rao and Z. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863. Google Scholar
[43]	S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations, 19 (1994), 277-319. doi: 10.1080/03605309408821017. Google Scholar
[44]	Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-46. Google Scholar
[45]	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. Google Scholar
[46]	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. Google Scholar
[47]	J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J., 28 (1979), 511-544. doi: 10.1512/iumj.1979.28.28037. Google Scholar
[48]	J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989. Google Scholar
[49]	S. Sugano, $L^p$ estimates for some Schrödinger type operators and a Calderón-Zygmund operator of Schrödinger type, Tokyo J. Math., 30 (2007), 179-197. doi: 10.3836/tjm/1184963655. Google Scholar
[50]	L. Tang, Weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators, preprint,, \arXiv{1109.0100}., (). Google Scholar
[51]	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. Google Scholar
[52]	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. Google Scholar
[53]	D. Yang and S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of $\mathbbR^n$, Rev. Mat. Iberoam., 29 (2013), 233-288. doi: 10.4171/RMI/719. Google Scholar
[54]	D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math., 55 (2012), 1677-1720. doi: 10.1007/s11425-012-4377-z. Google Scholar
[55]	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. Google Scholar
[56]	J. Zhong, The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line, 3 (1999), 1-48. Google Scholar

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, Ann. Inst. Fourier (Grenoble), 57 (2007), 1975-2013. Google Scholar
[2]	P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005. Google Scholar
[3]	P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248. doi: 10.1007/s12220-007-9003-x. Google Scholar
[4]	P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $\mathbbR^n$, J. Funct. Anal., 201 (2003), 148-184. doi: 10.1016/S0022-1236(03)00059-4. Google Scholar
[5]	N. Badr and B. Ben Ali, $L^p$ boundedness of the Riesz transform related to Schrödinger operators on a manifold, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8 (2009), 725-765. Google Scholar
[6]	A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358. doi: 10.5565/PUBLMAT_54210_03. Google Scholar
[7]	A. Bonami and S. Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces, Colloq. Math., 118 (2010), 107-132. doi: 10.4064/cm118-1-5. Google Scholar
[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, J. Math. Pures Appl. (9), 97 (2012), 230-241. doi: 10.1016/j.matpur.2011.06.002. Google Scholar
[9]	A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in $BMO$ and $H^1$, Ann. Inst. Fourier (Grenoble), 57 (2007), 1405-1439. Google Scholar
[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, Trans. Amer. Math. Soc., 365 (2013), 4729-4809. doi: 10.1090/S0002-9947-2013-05832-1. Google Scholar
[11]	R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), 72 (1993), 247-286. Google Scholar
[12]	R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304-335. doi: 10.1016/0022-1236(85)90007-2. Google Scholar
[13]	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. Google Scholar
[14]	L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657-700. doi: 10.1016/j.bulsci.2003.10.003. Google Scholar
[15]	L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal., 256 (2009), 1731-1768. doi: 10.1016/j.jfa.2009.01.017. Google Scholar
[16]	X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems, Rev. Mat. Iberoam., 29 (2013), 183-236. doi: 10.4171/RMI/718. Google Scholar
[17]	X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973. doi: 10.1090/S0894-0347-05-00496-0. Google Scholar
[18]	X. T. Duong and L. Yan, Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, J. Math. Soc. Japan, 63 (2011), 295-319. Google Scholar
[19]	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. Google Scholar
[20]	J. Dziubański and J. Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potential from reverse Hölder classes, Colloq. Math., 98 (2003), 5-38. doi: 10.4064/cm98-1-2. Google Scholar
[21]	C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. Google Scholar
[22]	J. García-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. (Rozprawy Mat.), 162 (1979), 1-63. Google Scholar
[23]	J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985. Google Scholar
[24]	F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. Google Scholar
[25]	L. Grafakos, Modern Fourier Analysis, 2nd edition, Graduate Texts in Mathematics 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2. Google Scholar
[26]	D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27-42. Google Scholar
[27]	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. Google Scholar
[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, Mem. Amer. Math. Soc., 214 (2011), no. 1007, vi+78 pp. doi: 10.1090/S0065-9266-2011-00624-6. Google Scholar
[29]	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. Google Scholar
[30]	S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4), 44 (2011), 723-800. Google Scholar
[31]	S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982. Google Scholar
[32]	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. Google Scholar
[33]	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. Google Scholar
[34]	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. Google Scholar
[35]	R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators, Sci. China Ser. A, 52 (2009), 1042-1080. doi: 10.1007/s11425-008-0136-6. Google Scholar
[36]	R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273. doi: 10.4171/RMI/50. Google Scholar
[37]	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. Google Scholar
[38]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983. Google Scholar
[39]	E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan, 37 (1985), 207-218. doi: 10.2969/jmsj/03720207. Google Scholar
[40]	E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005. Google Scholar
[41]	M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. Google Scholar
[42]	M. Rao and Z. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863. Google Scholar
[43]	S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations, 19 (1994), 277-319. doi: 10.1080/03605309408821017. Google Scholar
[44]	Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-46. Google Scholar
[45]	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. Google Scholar
[46]	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. Google Scholar
[47]	J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J., 28 (1979), 511-544. doi: 10.1512/iumj.1979.28.28037. Google Scholar
[48]	J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989. Google Scholar
[49]	S. Sugano, $L^p$ estimates for some Schrödinger type operators and a Calderón-Zygmund operator of Schrödinger type, Tokyo J. Math., 30 (2007), 179-197. doi: 10.3836/tjm/1184963655. Google Scholar
[50]	L. Tang, Weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators, preprint,, \arXiv{1109.0100}., (). Google Scholar
[51]	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. Google Scholar
[52]	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. Google Scholar
[53]	D. Yang and S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of $\mathbbR^n$, Rev. Mat. Iberoam., 29 (2013), 233-288. doi: 10.4171/RMI/719. Google Scholar
[54]	D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math., 55 (2012), 1677-1720. doi: 10.1007/s11425-012-4377-z. Google Scholar
[55]	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. Google Scholar
[56]	J. Zhong, The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line, 3 (1999), 1-48. Google Scholar

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2020 Impact Factor: 1.916

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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