Variable exponent Hardy spaces associated with discrete Laplacians on graphs
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摘要
In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.
In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.
In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.
引文
1 Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Comput Math Appl, 2008, 56:874-882
2 Acerbi E, Mingione G. Regularity results for a class of functionals with non-standard growth. Arch Ration Mech Anal,2001, 156:121-140
3 Adamowicz T, Harjulehto P, Hasto P. Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math Scand, 2015, 116:5-22
4 Alexopoulos G-K. Spectral multipliers on discrete groups. Bull Lond Math Soc, 2001, 33:417-424
5 Alexopoulos G-K. Spectral multipliers for Markov chains. J Math Soc Japan, 2004, 56:833-852
6 Badr N, Martell J-M. Weighted norm inequalities on graphs. J Geom Anal, 2012, 22:1173-1210
7 Badr N, Russ E. Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs. Publ Mat, 2009, 53:273-328
8 Bui T-A. Weighted Hardy spaces associated to discrete Laplacians on graphs and applications. Potential Anal, 2014,41:817-848
9 Bui T-A, Cao J, Ky L-D, et al. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced offdiagonal estimates. Anal Geom Metr Spaces, 2013, 1:69-129
10 Bui T-A, Duong X-T. Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals. Trans Amer Math Soc, 2014, 366:3451-3485
11 Bui T-A, Duong X-T. Weighted norm inequalities for spectral multipliers on graphs. Potential Anal, 2016, 44:263-293
12 Bui T-A, Duong X-T, Ly F-K. Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type. Trans Amer Math Soc, 2018, 370:7229-7292
13 Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math,2006, 66:1383-1406
14 Coifman R-R, Meyer Y, Stein E-M. Some new function spaces and their applications to harmonic analysis. J Funct Anal, 1985, 62:304-335
15 Coifman R-R, Weiss G. Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Lecture Notes in Mathematics, vol. 242. Berlin:Springer-Verlag, 1971
16 Coifman R-R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83:569-645
17 Cowling M, Meda S, Setti A-G. Estimates for functions of the Laplace operator on homogeneous trees. Trans Amer Math Soc, 2000, 352:4271-4293
18 Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Heidelberg:Birkhauser/Springer, 2013
19 Cruz-Uribe D, Fiorenza A, Martell J-M, et al. The boundedness of classical operators on variable L~P spaces. Ann Acad Sci Fenn Math, 2006, 31:239-264
20 Cruz-Uribe D, Fiorenza A, Neugebauer C-J. Corrections to:"The maximal function on variable L~p spaces". Ann Acad Sci Fenn Math, 2003, 28:223-238
21 Cruz-Uribe D, Wang LAD. Variable Hardy spaces. Indiana Univ Math J, 2014, 63:447-493
22 Diening L. Theoretical and numerical results for electrorheological fluids. PhD Thesis. Germany:University of Freiburg, 1998
23 Diening L. Maximal function on generalized Lebesgue spaces L~(p(·)).Math Inequal Appl, 2004, 7:245-253
24 Diening L, Harjulehto P, Hasto P, et al. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Heidelberg:Springer, 2011
25 Dungey N. A note on time regularity for discrete time heat kernels. Semigroup Forum, 2006, 72:404-410
26 Dziubanski J, Preisner M. Hardy spaces for semigroups with Gaussian bounds. Ann Mat Pura Appl(4), 2018, 197:965-987
27 Feneuil J. Littlewood-Paley functionals on graphs. Math Nachr, 2015, 288:1254-1285
28 Feneuil J. Hardy and BMO spaces on graphs, application to Riesz transform. Potential Anal, 2016, 45:1-54
29 Grafakos L, Liu L, Yang D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications.Sci China Ser A, 2008, 51:2253-2284
30 Grafakos L, Liu L, Yang D. Radial maximal function characterizations for Hardy spaces on RD-spaces. Bull Soc Math France, 2009, 137:225-251
31 Han Y, Miiller D, Yang D. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Caratheodory spaces. Abstr Appl Anal, 2008, 2008:893409
32 Harjulehto P, Hasto P, Pere M. Variable exponent Lebesgue spaces on metric spaces:The Hardy-Littlewood maximal operator. Real Anal Exchange, 2004, 30:87-103
33 Khabazi M. The maximal operator in spaces of homogenous type. Proc A Razmadze Math Inst, 2005, 138:17-25
34 Kokilashvili V, Samko S. The maximal operator in weighted variable spaces on metric measure spaces. Proc A Razmadze Math Inst, 2007, 144:137-144
35 Kovacik O, Rakosnik J-M. On spaces L~(p(x))and W~(k,p(x)). Czechoslovak Math J, 1991, 41:592-618
36 Kyrezi I, Marias M. H~p-bounds for spectral multipliers on graphs. Trans Amer Math Soc, 2009, 361:1053-1067
37 Macias R-A, Segovia C. A decomposition into atoms of distributions on spaces of homogeneous type. Adv Math, 1979,33:271-309
38 Macias R-A, Segovia C. Lipschitz functions on spaces of homogeneous type. Adv Math, 1979, 33:257-270
39 Martell J-M, Prisuelos-Arribas C. Weighted norm inequalities for conical square functions. Part I. Trans Amer Math Soc, 2017, 369:4193-4233
40 Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012,262:3665-3748
41 Nakai E, Yabuta K. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math Japon, 1997, 46:15-28
42 Nakano H. Modulared Semi-Ordered Linear Spaces. Tokyo:Maruzen, 1950
43 Nekvinda A. A note on maximal operator on l~({p_n})and L~(p(x))(R). J Funct Spaces Appl, 2007, 5:49-88
44 Orlicz W. Uber konjugierte exponentenfolgen. Studia Math, 1931, 3:200-211
45 Pang M-M-H. Heat kernels of graphs. J Lond Math Soc, 1993, 47:50-64
46 Pang M-M-H. The heat kernel of the Laplacian defined on a uniform grid. Semigroup Forum, 2009, 78:238-252
47 Pick L-S, Ruzicka M. An example of a space L~(p(x))on which the Hardy-Littlewood maximal operator is not bounded.Expo Math, 2001, 19:369-371
48 Rajagopal K-R, Ruzicka M. On the modeling of electrorheological materials. Mech Res Comm, 1996, 23:401-407
49 Russ E. Espaces de Hardy et transformees de Riesz sur les graphes et les varietes. PhD Thesis. Pontoise:University Cergy, 1998
50 Russ E. H~1-BMO duality on graphs. Colloq Math, 2000, 86:67-91
51 Russ E. Riesz transforms on graphs for 1≤p≤2. Math Scand, 2000, 87:133-160
52 Russ E. The atomic decomposition for tent spaces on spaces of homogeneous type. In:CMA/AMSI Research Symposium"Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics". Proceedings of the Centre for Mathematics and its Applications, vol. 42. Canberra:The Australian National University, 2007, 125-135
53 Ruzicka M. Electrorheological Fluids:Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748.Berlin:Springer-Verlag, 2000
54 Sawano Y. Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equations Operator Theory, 2013, 77:123-148
55 Song L, Yan L. Maximal function characterizations for Hardy spaces associated with non-negative self-adjoint operators on spaces of homogeneous type. J Evol Equ, 2018, 18:221-243
56 Uchiyama A. A maximal function characterization of Hp on the space of homogeneous type. Trans Amer Math Soc,1980, 262:579-592
57 Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc,2008, 360:4383-4408
58 Yang D, Yang S. Musielak-Orlicz-Hardy spaces associated with operators and their applications. J Geom Anal, 2014,24:495-570
59 Yang D, Yang S. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative selfadjoint operators satisfying Gaussian estimates. Commun Pure Appl Anal, 2016, 15:2135-2160
60 Yang D, Zhang J, Zhuo C. Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Proc Edinb Math Soc(2), arXiv:1601.06358, 2016
61 Yang D, Zhou Y. Localized Hardy spaces H1 related to admisible functions on RD-spaces and applications to Schrodinger operators. Trans Amer Math Soc, 2011, 363:1197-1239
62 Yang D, Zhuo C. Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators.Ann Acad Sci Fenn Math, 2016, 41:357-398
63 Yang D, Zhuo C, Nakai E. Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev Mat Complut, 2016, 29:245-270
64 Zhuo C, Sawano Y, Yang D. Hardy spaces with variable exponents on RD-spaces and applications. Dissertations Math(Rozprawy Mat), 2016, 520:1-74
65 Zhuo C, Yang D. Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates. Nonlinear Anal, 2016, 141:16-42
66 Zhuo C, Yang D, Liang Y. Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull Malays Math Sci Soc(2), 2016, 39:1541-1577
2 Acerbi E, Mingione G. Regularity results for a class of functionals with non-standard growth. Arch Ration Mech Anal,2001, 156:121-140
3 Adamowicz T, Harjulehto P, Hasto P. Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math Scand, 2015, 116:5-22
4 Alexopoulos G-K. Spectral multipliers on discrete groups. Bull Lond Math Soc, 2001, 33:417-424
5 Alexopoulos G-K. Spectral multipliers for Markov chains. J Math Soc Japan, 2004, 56:833-852
6 Badr N, Martell J-M. Weighted norm inequalities on graphs. J Geom Anal, 2012, 22:1173-1210
7 Badr N, Russ E. Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs. Publ Mat, 2009, 53:273-328
8 Bui T-A. Weighted Hardy spaces associated to discrete Laplacians on graphs and applications. Potential Anal, 2014,41:817-848
9 Bui T-A, Cao J, Ky L-D, et al. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced offdiagonal estimates. Anal Geom Metr Spaces, 2013, 1:69-129
10 Bui T-A, Duong X-T. Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals. Trans Amer Math Soc, 2014, 366:3451-3485
11 Bui T-A, Duong X-T. Weighted norm inequalities for spectral multipliers on graphs. Potential Anal, 2016, 44:263-293
12 Bui T-A, Duong X-T, Ly F-K. Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type. Trans Amer Math Soc, 2018, 370:7229-7292
13 Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math,2006, 66:1383-1406
14 Coifman R-R, Meyer Y, Stein E-M. Some new function spaces and their applications to harmonic analysis. J Funct Anal, 1985, 62:304-335
15 Coifman R-R, Weiss G. Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Lecture Notes in Mathematics, vol. 242. Berlin:Springer-Verlag, 1971
16 Coifman R-R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83:569-645
17 Cowling M, Meda S, Setti A-G. Estimates for functions of the Laplace operator on homogeneous trees. Trans Amer Math Soc, 2000, 352:4271-4293
18 Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Heidelberg:Birkhauser/Springer, 2013
19 Cruz-Uribe D, Fiorenza A, Martell J-M, et al. The boundedness of classical operators on variable L~P spaces. Ann Acad Sci Fenn Math, 2006, 31:239-264
20 Cruz-Uribe D, Fiorenza A, Neugebauer C-J. Corrections to:"The maximal function on variable L~p spaces". Ann Acad Sci Fenn Math, 2003, 28:223-238
21 Cruz-Uribe D, Wang LAD. Variable Hardy spaces. Indiana Univ Math J, 2014, 63:447-493
22 Diening L. Theoretical and numerical results for electrorheological fluids. PhD Thesis. Germany:University of Freiburg, 1998
23 Diening L. Maximal function on generalized Lebesgue spaces L~(p(·)).Math Inequal Appl, 2004, 7:245-253
24 Diening L, Harjulehto P, Hasto P, et al. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Heidelberg:Springer, 2011
25 Dungey N. A note on time regularity for discrete time heat kernels. Semigroup Forum, 2006, 72:404-410
26 Dziubanski J, Preisner M. Hardy spaces for semigroups with Gaussian bounds. Ann Mat Pura Appl(4), 2018, 197:965-987
27 Feneuil J. Littlewood-Paley functionals on graphs. Math Nachr, 2015, 288:1254-1285
28 Feneuil J. Hardy and BMO spaces on graphs, application to Riesz transform. Potential Anal, 2016, 45:1-54
29 Grafakos L, Liu L, Yang D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications.Sci China Ser A, 2008, 51:2253-2284
30 Grafakos L, Liu L, Yang D. Radial maximal function characterizations for Hardy spaces on RD-spaces. Bull Soc Math France, 2009, 137:225-251
31 Han Y, Miiller D, Yang D. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Caratheodory spaces. Abstr Appl Anal, 2008, 2008:893409
32 Harjulehto P, Hasto P, Pere M. Variable exponent Lebesgue spaces on metric spaces:The Hardy-Littlewood maximal operator. Real Anal Exchange, 2004, 30:87-103
33 Khabazi M. The maximal operator in spaces of homogenous type. Proc A Razmadze Math Inst, 2005, 138:17-25
34 Kokilashvili V, Samko S. The maximal operator in weighted variable spaces on metric measure spaces. Proc A Razmadze Math Inst, 2007, 144:137-144
35 Kovacik O, Rakosnik J-M. On spaces L~(p(x))and W~(k,p(x)). Czechoslovak Math J, 1991, 41:592-618
36 Kyrezi I, Marias M. H~p-bounds for spectral multipliers on graphs. Trans Amer Math Soc, 2009, 361:1053-1067
37 Macias R-A, Segovia C. A decomposition into atoms of distributions on spaces of homogeneous type. Adv Math, 1979,33:271-309
38 Macias R-A, Segovia C. Lipschitz functions on spaces of homogeneous type. Adv Math, 1979, 33:257-270
39 Martell J-M, Prisuelos-Arribas C. Weighted norm inequalities for conical square functions. Part I. Trans Amer Math Soc, 2017, 369:4193-4233
40 Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012,262:3665-3748
41 Nakai E, Yabuta K. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math Japon, 1997, 46:15-28
42 Nakano H. Modulared Semi-Ordered Linear Spaces. Tokyo:Maruzen, 1950
43 Nekvinda A. A note on maximal operator on l~({p_n})and L~(p(x))(R). J Funct Spaces Appl, 2007, 5:49-88
44 Orlicz W. Uber konjugierte exponentenfolgen. Studia Math, 1931, 3:200-211
45 Pang M-M-H. Heat kernels of graphs. J Lond Math Soc, 1993, 47:50-64
46 Pang M-M-H. The heat kernel of the Laplacian defined on a uniform grid. Semigroup Forum, 2009, 78:238-252
47 Pick L-S, Ruzicka M. An example of a space L~(p(x))on which the Hardy-Littlewood maximal operator is not bounded.Expo Math, 2001, 19:369-371
48 Rajagopal K-R, Ruzicka M. On the modeling of electrorheological materials. Mech Res Comm, 1996, 23:401-407
49 Russ E. Espaces de Hardy et transformees de Riesz sur les graphes et les varietes. PhD Thesis. Pontoise:University Cergy, 1998
50 Russ E. H~1-BMO duality on graphs. Colloq Math, 2000, 86:67-91
51 Russ E. Riesz transforms on graphs for 1≤p≤2. Math Scand, 2000, 87:133-160
52 Russ E. The atomic decomposition for tent spaces on spaces of homogeneous type. In:CMA/AMSI Research Symposium"Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics". Proceedings of the Centre for Mathematics and its Applications, vol. 42. Canberra:The Australian National University, 2007, 125-135
53 Ruzicka M. Electrorheological Fluids:Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748.Berlin:Springer-Verlag, 2000
54 Sawano Y. Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equations Operator Theory, 2013, 77:123-148
55 Song L, Yan L. Maximal function characterizations for Hardy spaces associated with non-negative self-adjoint operators on spaces of homogeneous type. J Evol Equ, 2018, 18:221-243
56 Uchiyama A. A maximal function characterization of Hp on the space of homogeneous type. Trans Amer Math Soc,1980, 262:579-592
57 Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc,2008, 360:4383-4408
58 Yang D, Yang S. Musielak-Orlicz-Hardy spaces associated with operators and their applications. J Geom Anal, 2014,24:495-570
59 Yang D, Yang S. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative selfadjoint operators satisfying Gaussian estimates. Commun Pure Appl Anal, 2016, 15:2135-2160
60 Yang D, Zhang J, Zhuo C. Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Proc Edinb Math Soc(2), arXiv:1601.06358, 2016
61 Yang D, Zhou Y. Localized Hardy spaces H1 related to admisible functions on RD-spaces and applications to Schrodinger operators. Trans Amer Math Soc, 2011, 363:1197-1239
62 Yang D, Zhuo C. Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators.Ann Acad Sci Fenn Math, 2016, 41:357-398
63 Yang D, Zhuo C, Nakai E. Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev Mat Complut, 2016, 29:245-270
64 Zhuo C, Sawano Y, Yang D. Hardy spaces with variable exponents on RD-spaces and applications. Dissertations Math(Rozprawy Mat), 2016, 520:1-74
65 Zhuo C, Yang D. Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates. Nonlinear Anal, 2016, 141:16-42
66 Zhuo C, Yang D, Liang Y. Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull Malays Math Sci Soc(2), 2016, 39:1541-1577