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Some properties of positive solutions for an integral system with the double weighted Riesz potentials
Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates
1. | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875 |
2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005. |
[2] |
P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $R^n$, J. Funct. Anal., 201 (2003), 148-184.
doi: 10.1016/S0022-1236(03)00059-4. |
[3] |
P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque, 249 (1998), viii+172 pp. |
[4] |
T. A. Bui, J. Cao, L. D. Ky, D. Yang and S. Yang, Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129. |
[5] |
F. Cacciafesta and P. D'Ancona, Weighted $L^p$ estimates for powers of selfadjoint operators, Adv. Math., 229 (2012), 501-530.
doi: 10.1016/j.aim.2011.09.007. |
[6] |
A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces, Adv. Math., 25 (1977), 216-225. |
[7] |
J. Cao, D.-C. Chang, D. Yang and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463.
doi: 10.3934/cpaa.2014.13.1435. |
[8] |
J. Cao, S. Mayboroda and D. Yang, Local Hardy spaces associated with inhomogeneous higher order elliptic operators, Anal. Appl. (Singap.), (2015).
doi: 10.1142/S0219530515500189. |
[9] |
J. Cao, S. Mayboroda and D. Yang, Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators,, \emph{Forum Math.}, ().
doi: 10.1515/forum-2014-0127. |
[10] |
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. |
[11] |
X. T. Duong and J. Li, Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal., 264 (2013), 1409-1437.
doi: 10.1016/j.jfa.2013.01.006. |
[12] |
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. |
[13] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. |
[14] |
L. Grafakos, Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics 250, Springer, New York, 2009.
doi: 10.1007/978-0-387-09434-2. |
[15] |
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. |
[16] |
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. |
[17] |
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.
doi: 10.1142/S0219199713500296. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273.
doi: 10.4171/RMI/50. |
[22] |
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. |
[23] |
L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958.
doi: 10.1090/S0002-9947-2012-05727-8. |
[24] |
S. Liu and L. Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators, J. Funct. Anal., 265 (2013), 2709-2723.
doi: 10.1016/j.jfa.2013.08.003. |
[25] |
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983.
doi: 10.1007/BFb0072210. |
[26] |
M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.
doi: 10.1080/03601239109372748. |
[27] |
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. |
[28] |
L. Song and L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math., 287 (2016), 463-484.
doi: 10.1016/j.aim.2015.09.026. |
[29] |
E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993. |
[30] |
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. |
[31] |
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. |
[32] |
Da. Yang and Do. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators, Front. Math. China, 10 (2015), 1203-1232.
doi: 10.1007/s11464-015-0432-8. |
[33] |
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. |
[34] |
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. |
show all references
References:
[1] |
P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005. |
[2] |
P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $R^n$, J. Funct. Anal., 201 (2003), 148-184.
doi: 10.1016/S0022-1236(03)00059-4. |
[3] |
P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque, 249 (1998), viii+172 pp. |
[4] |
T. A. Bui, J. Cao, L. D. Ky, D. Yang and S. Yang, Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129. |
[5] |
F. Cacciafesta and P. D'Ancona, Weighted $L^p$ estimates for powers of selfadjoint operators, Adv. Math., 229 (2012), 501-530.
doi: 10.1016/j.aim.2011.09.007. |
[6] |
A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces, Adv. Math., 25 (1977), 216-225. |
[7] |
J. Cao, D.-C. Chang, D. Yang and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463.
doi: 10.3934/cpaa.2014.13.1435. |
[8] |
J. Cao, S. Mayboroda and D. Yang, Local Hardy spaces associated with inhomogeneous higher order elliptic operators, Anal. Appl. (Singap.), (2015).
doi: 10.1142/S0219530515500189. |
[9] |
J. Cao, S. Mayboroda and D. Yang, Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators,, \emph{Forum Math.}, ().
doi: 10.1515/forum-2014-0127. |
[10] |
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. |
[11] |
X. T. Duong and J. Li, Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal., 264 (2013), 1409-1437.
doi: 10.1016/j.jfa.2013.01.006. |
[12] |
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. |
[13] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. |
[14] |
L. Grafakos, Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics 250, Springer, New York, 2009.
doi: 10.1007/978-0-387-09434-2. |
[15] |
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. |
[16] |
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. |
[17] |
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.
doi: 10.1142/S0219199713500296. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoam., 3 (1987), 249-273.
doi: 10.4171/RMI/50. |
[22] |
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. |
[23] |
L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958.
doi: 10.1090/S0002-9947-2012-05727-8. |
[24] |
S. Liu and L. Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators, J. Funct. Anal., 265 (2013), 2709-2723.
doi: 10.1016/j.jfa.2013.08.003. |
[25] |
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983.
doi: 10.1007/BFb0072210. |
[26] |
M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.
doi: 10.1080/03601239109372748. |
[27] |
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. |
[28] |
L. Song and L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math., 287 (2016), 463-484.
doi: 10.1016/j.aim.2015.09.026. |
[29] |
E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993. |
[30] |
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. |
[31] |
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. |
[32] |
Da. Yang and Do. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators, Front. Math. China, 10 (2015), 1203-1232.
doi: 10.1007/s11464-015-0432-8. |
[33] |
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. |
[34] |
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. |
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