# American Institute of Mathematical Sciences

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

 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:
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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. 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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. 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