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Green's functions for parabolic systems of second order in time-varying domains
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 |
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. |
[2] |
P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. |
[22] |
J. García-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. (Rozprawy Mat.), 162 (1979), 1-63. |
[23] |
J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985. |
[24] |
F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. |
[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. |
[26] |
D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27-42. |
[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. |
[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. |
[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. |
[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. |
[31] |
S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982. |
[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. |
[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. |
[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. |
[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. |
[36] |
R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273.
doi: 10.4171/RMI/50. |
[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. |
[38] |
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983. |
[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. |
[40] |
E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005. |
[41] |
M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. |
[42] |
M. Rao and Z. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002.
doi: 10.1201/9780203910863. |
[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. |
[44] |
Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-46. |
[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. |
[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. |
[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. |
[48] |
J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989. |
[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. |
[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, Trans. Amer. Math. Soc., 360 (2008), 4383-4408.
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 $\mathbbR^n$, Indiana Univ. Math. J., 61 (2012), 81-129.
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 $\mathbbR^n$, Rev. Mat. Iberoam., 29 (2013), 233-288.
doi: 10.4171/RMI/719. |
[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. |
[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. |
[56] |
J. Zhong, The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line, 3 (1999), 1-48. |
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. |
[2] |
P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. |
[22] |
J. García-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. (Rozprawy Mat.), 162 (1979), 1-63. |
[23] |
J. Garc\ía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985. |
[24] |
F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. |
[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. |
[26] |
D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27-42. |
[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. |
[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. |
[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. |
[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. |
[31] |
S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982. |
[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. |
[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. |
[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. |
[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. |
[36] |
R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273.
doi: 10.4171/RMI/50. |
[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. |
[38] |
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983. |
[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. |
[40] |
E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005. |
[41] |
M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. |
[42] |
M. Rao and Z. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002.
doi: 10.1201/9780203910863. |
[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. |
[44] |
Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-46. |
[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. |
[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. |
[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. |
[48] |
J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989. |
[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. |
[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, Trans. Amer. Math. Soc., 360 (2008), 4383-4408.
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 $\mathbbR^n$, Indiana Univ. Math. J., 61 (2012), 81-129.
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 $\mathbbR^n$, Rev. Mat. Iberoam., 29 (2013), 233-288.
doi: 10.4171/RMI/719. |
[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. |
[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. |
[56] |
J. Zhong, The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line, 3 (1999), 1-48. |
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