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:
 [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. Google Scholar [2] 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 [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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,, \emph{J. Math. Pures Appl. (9)}, 97 (2012), 230. 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 $H^1$,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1405. 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,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 4729. 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,, \emph{J. Math. Pures Appl. (9)}, 72 (1993), 247. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 183. 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,, \emph{J. Amer. Math. Soc.}, 18 (2005), 943. 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,, \emph{J. Math. Soc. Japan}, 63 (2011), 295. Google Scholar [19] J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45. Google Scholar [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. Google Scholar [21] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137. Google Scholar [22] J. García-Cuerva, Weighted $H^p$ spaces,, \emph{Dissertationes Math. (Rozprawy Mat.)}, 162 (1979), 1. Google Scholar [23] J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics,, Amsterdam, (1985). Google Scholar [24] F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping,, \emph{Acta Math.}, 130 (1973), 265. Google Scholar [25] L. Grafakos, Modern Fourier Analysis,, 2$^{nd}$ edition, (2009). doi: 10.1007/978-0-387-09434-2. Google Scholar [26] D. Goldberg, A local version of real Hardy spaces,, \emph{Duke Math. J.}, 46 (1979), 27. Google Scholar [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). 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,, \emph{Mem. Amer. Math. Soc.}, 214 (2011). 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,, \emph{Math. Ann.}, 344 (2009), 37. 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,, \emph{Ann. Sci. \'Ecole Norm. Sup. (4)}, 44 (2011), 723. Google Scholar [31] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation,, \emph{Duke Math. J.}, 47 (1980), 959. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$,, \emph{Rev. Mat. Iberoam.}, 3 (1987), 249. doi: 10.4171/RMI/50. Google Scholar [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. Google Scholar [38] J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math., (1034). Google Scholar [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. Google Scholar [40] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005). Google Scholar [41] M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991). Google Scholar [42] M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker, (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,, \emph{Comm. Partial Differential Equations}, 19 (1994), 277. doi: 10.1080/03605309408821017. Google Scholar [44] Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential,, \emph{Ann. Inst. Fourier (Grenoble)}, 45 (1995), 513. Google Scholar [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. 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,, \emph{Acta Math.}, 103 (1960), 25. Google Scholar [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. Google Scholar [48] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces,, Lecture Notes in Math., (1381). Google Scholar [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. 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,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 4383. 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 $\mathbbR^n$,, \emph{Indiana Univ. Math. J.}, 61 (2012), 81. 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 $\mathbbR^n$,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 233. doi: 10.4171/RMI/719. Google Scholar [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. Google Scholar [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. Google Scholar [56] J. Zhong, The Sobolev estimates for some Schrödinger type operators,, \emph{Math. Sci. Res. Hot-Line}, 3 (1999), 1. 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,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1975. Google Scholar [2] 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 [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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,, \emph{J. Math. Pures Appl. (9)}, 97 (2012), 230. 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 $H^1$,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 1405. 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,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 4729. 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,, \emph{J. Math. Pures Appl. (9)}, 72 (1993), 247. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 183. 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,, \emph{J. Amer. Math. Soc.}, 18 (2005), 943. 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,, \emph{J. Math. Soc. Japan}, 63 (2011), 295. Google Scholar [19] J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators,, in \emph{Fourier analysis and related topics, (2002), 45. Google Scholar [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. Google Scholar [21] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, \emph{Acta Math.}, 129 (1972), 137. Google Scholar [22] J. García-Cuerva, Weighted $H^p$ spaces,, \emph{Dissertationes Math. (Rozprawy Mat.)}, 162 (1979), 1. Google Scholar [23] J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics,, Amsterdam, (1985). Google Scholar [24] F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping,, \emph{Acta Math.}, 130 (1973), 265. Google Scholar [25] L. Grafakos, Modern Fourier Analysis,, 2$^{nd}$ edition, (2009). doi: 10.1007/978-0-387-09434-2. Google Scholar [26] D. Goldberg, A local version of real Hardy spaces,, \emph{Duke Math. J.}, 46 (1979), 27. Google Scholar [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). 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,, \emph{Mem. Amer. Math. Soc.}, 214 (2011). 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,, \emph{Math. Ann.}, 344 (2009), 37. 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,, \emph{Ann. Sci. \'Ecole Norm. Sup. (4)}, 44 (2011), 723. Google Scholar [31] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation,, \emph{Duke Math. J.}, 47 (1980), 959. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$,, \emph{Rev. Mat. Iberoam.}, 3 (1987), 249. doi: 10.4171/RMI/50. Google Scholar [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. Google Scholar [38] J. Musielak, Orlicz Spaces and Modular Spaces,, Lecture Notes in Math., (1034). Google Scholar [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. Google Scholar [40] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005). Google Scholar [41] M. Rao and Z. Ren, Theory of Orlicz Spaces,, Marcel Dekker, (1991). Google Scholar [42] M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker, (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,, \emph{Comm. Partial Differential Equations}, 19 (1994), 277. doi: 10.1080/03605309408821017. Google Scholar [44] Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential,, \emph{Ann. Inst. Fourier (Grenoble)}, 45 (1995), 513. Google Scholar [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. 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,, \emph{Acta Math.}, 103 (1960), 25. Google Scholar [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. Google Scholar [48] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces,, Lecture Notes in Math., (1381). Google Scholar [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. 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,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 4383. 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 $\mathbbR^n$,, \emph{Indiana Univ. Math. J.}, 61 (2012), 81. 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 $\mathbbR^n$,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 233. doi: 10.4171/RMI/719. Google Scholar [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. Google Scholar [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. Google Scholar [56] J. Zhong, The Sobolev estimates for some Schrödinger type operators,, \emph{Math. Sci. Res. Hot-Line}, 3 (1999), 1. Google Scholar
 [1] 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 [2] 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 [3] Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169 [4] Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831 [5] 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 [6] Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174 [7] Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541 [8] Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037 [9] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [10] Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure & Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533 [11] Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025 [12] Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377 [13] Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933 [14] Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013 [15] Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001 [16] Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59 [17] Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems & Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055 [18] Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143 [19] Elvise Berchio, Debdip Ganguly. Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020 [20] Florian Schneider. Second-order mixed-moment model with differentiable ansatz function in slab geometry. Kinetic & Related Models, 2018, 11 (5) : 1255-1276. doi: 10.3934/krm.2018049

2018 Impact Factor: 0.925