Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrdinger operators
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摘要
Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.
Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.
Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.
引文
1 Bernardis A L, Pradolini G, Lorente M, et al. Composition of fractional Orlicz maximal operators and A 1 -weights on spaces of homogeneous type. Acta Math Sinica Engl Ser, 2010, 8: 1509–1518
2 Bongioanni B, Harboure E, Salinas O. Commutators of Riesz transforms related to Schrdinger operators. J Fourier Anal Appl, 2011, 17: 115–134
3 Bramanti M, Cerutti M. Commutators of singular integrals on homogeneous spaces. Boll Un Mat ltd, 1996, 10: 843–883
4 Bramanti M, Brandolini L. Estimates of BMO type for singular integrals on spaces of homogeneous type and applica- tions to hypoelliptic PDEs. Rev Mat Iberoamericana, 2005, 21: 511–556
5 Burger N. Espace des fonctions à variation moyenne bornée sur un espace de nature homogène. C R Acad Sci Paris Sr, 1978, 286: 139–142
6 Cao J, Liu Y, Yang D C. Hardy spaces H1 L (R n ) associated to Schrdinger type operators. Houston J Math, 2010, 36: 1067–1095
7 Cao J, Yang D C. Hardy spaces H p L (R n ) associated with operators satisfying k -Davies-Gaffney estimates. Sci China Math, 2012, 55: 1403–1440
8 Coifman R R, Weiss G. Analyse Harmonique Non-Commutative sur Certain Espaces Homogènes. Lecture Notes in Math 242. Berlin: Springer-Verlag, 1971
9 Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645
10 Coifman R R, Rochberg R, Weiss G. Factorization theorem for Hardy spaces in several variables. Ann Math, 1976,103: 611–635
11 Duong X T, Yan L X. Commutators of BMO functions and singular integral operators with non-smooth kernels. Bull Austral Math Soc, 2003, 67: 187–200
12 Duong X T, Yan L X. Duality of Hardy and BMO spaces associated with operators with heat kernel. J Amer Math Soc, 2005, 18: 943–973
13 Dziubański J, Zienkiewicz J. Hardy space H1 associated to Schrdinger operator with potential satisfying reverse Hlder inequality. Rev Mat Iberoamericana, 1999, 15: 279–296
14 Dziubański J, Garrigos G, Martinez T, et al. BMO spaces related to Schrdinger operators with potentials satisfying a reverse Hlder inequality. Math Z, 2005, 249: 329–356
15 Guo Z H, Li P T, Peng L Z. L p boundedness of commutators of Riesz transform associated to Schrdinger operator. J Math Anal Appl, 2008, 341: 421–432
16 Han Y, Mller D, Yang D C. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 2006, 279: 1505–1537
17 Han Y, Mller D, Yang D C. A theory of Besov and Triebel-Lizorkin spaces on metric measure space modeled on Carnot-Carathéodory spaces. Abstr Appl Anal, 2008, Art. ID 893409, 250 pp
18 Hofmann S, Lu G, Mitrea D, et al. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies- Gaffney estimates. Mem Amer Math Soc, 2011, 1007: 1–78
19 Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344: 37–116
20 Janson S. Mean oscillation and commutators of singular integrals operators. Ark Math, 1978, 16: 263–270
21 Jiang R, Yang D C. Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun Contemp Math, 2011, 13: 331–373
22 Jiang R, Yang D C. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167–1224
23 Jiang R, Yang D C. Generalized vanishing mean oscillation spaces associated with divergence form elliptic operators. Integral Equations Operator Theory, 2010, 67: 123–149
24 Jiang R, Yang D C, Zhou Y. Orlicz-Hardy spaces associated with operators. Sci China Ser A, 2009, 52: 1042–1080
25 Li H. Estimations L p des opéateurs de Schringer sur les groupes nilpotents. J Funct Anal, 1999, 161: 152–218
26 Li P T, Peng L Z. Enpoint estimates for commutators of Riesz transforms associated with Schrdinger operators. Bull Aust Math Soc, 2010, 82: 367–389
27 Liang Y, Yang D C, Yang S. Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates. Sci China Math, 2011, 54: 2395–2426
28 Lin C C, Liu H P, Liu Y. Hardy spaces associated with Schrdinger operators on the Heisenberg group. ArXiv:1106.4960
29 Lin C C, Liu H P. BM O L (H n ) spaces and Carleson measures for Schrdinger operators. Adv Math, 2011, 228: 1631–1688
30 Liu Y, Dong J F. Some estimates of higher order Riesz transform related to Schrdinger type operator. Potential Anal, 2010, 32: 41–55
31 Lu G Z. Embedding theorem into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ Mat, 1996, 40: 301–329
32 Lu S Z, Wu Q, Yang D C. Boundedness of commutators on Hardy type space. Sci China Ser A, 2002, 45: 984-997
33 Macías R, Segovia C. A well-behaved quasi-distance on spaces of homogeneous type. Trab Mat Inst Argentina Mat, 1981, 32: 1–18
34 Nagel A, Stein E M, Wainger S. Balls and metrics defined by vector fields I. Basic properties. Acta Math, 1985, 155: 103–147
35 Pradolini G, Salinas O. Commutators of singular integrals on spaces of homogeneous type. Czechoslov Math J, 2007,57: 75–93
36 Shen Z W. L p -estimates for Schrdinger operators with certain potentials. Ann Inst Fourier Grenoble, 1995, 45: 513–546
37 Sun Y Z, Su W Y. Hardy type estimates for commutators of singular integrals. Sci China Ser A, 2005, 48: 563-575
38 Yan L X. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 2008, 360: 4383–4408
39 Yang D C, Zhou Y. Localized Hardy spaces H1 related to admissible functions on RD-spaces and applications to Schrdinger operators. Trans Amer Math Soc, 2011, 363: 1197–1239
40 Yang D C, Yang D Y, Zhou Y. Localized BMO and BLO spaces on RD-spaces and applications to Schrdinger operators. Commun Pure Appl Anal, 2010, 9: 779–812
41 Yang D C, Yang D Y, Zhou Y. Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrdinger operators. Nagoya Math J, 2010, 198: 77–119
42 Yang D C, Yang S B. Local Hardy spaces of Musielak-Orlicz type and their applications. Sci China Math, 2012, 55: 1677–1720
2 Bongioanni B, Harboure E, Salinas O. Commutators of Riesz transforms related to Schrdinger operators. J Fourier Anal Appl, 2011, 17: 115–134
3 Bramanti M, Cerutti M. Commutators of singular integrals on homogeneous spaces. Boll Un Mat ltd, 1996, 10: 843–883
4 Bramanti M, Brandolini L. Estimates of BMO type for singular integrals on spaces of homogeneous type and applica- tions to hypoelliptic PDEs. Rev Mat Iberoamericana, 2005, 21: 511–556
5 Burger N. Espace des fonctions à variation moyenne bornée sur un espace de nature homogène. C R Acad Sci Paris Sr, 1978, 286: 139–142
6 Cao J, Liu Y, Yang D C. Hardy spaces H1 L (R n ) associated to Schrdinger type operators. Houston J Math, 2010, 36: 1067–1095
7 Cao J, Yang D C. Hardy spaces H p L (R n ) associated with operators satisfying k -Davies-Gaffney estimates. Sci China Math, 2012, 55: 1403–1440
8 Coifman R R, Weiss G. Analyse Harmonique Non-Commutative sur Certain Espaces Homogènes. Lecture Notes in Math 242. Berlin: Springer-Verlag, 1971
9 Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645
10 Coifman R R, Rochberg R, Weiss G. Factorization theorem for Hardy spaces in several variables. Ann Math, 1976,103: 611–635
11 Duong X T, Yan L X. Commutators of BMO functions and singular integral operators with non-smooth kernels. Bull Austral Math Soc, 2003, 67: 187–200
12 Duong X T, Yan L X. Duality of Hardy and BMO spaces associated with operators with heat kernel. J Amer Math Soc, 2005, 18: 943–973
13 Dziubański J, Zienkiewicz J. Hardy space H1 associated to Schrdinger operator with potential satisfying reverse Hlder inequality. Rev Mat Iberoamericana, 1999, 15: 279–296
14 Dziubański J, Garrigos G, Martinez T, et al. BMO spaces related to Schrdinger operators with potentials satisfying a reverse Hlder inequality. Math Z, 2005, 249: 329–356
15 Guo Z H, Li P T, Peng L Z. L p boundedness of commutators of Riesz transform associated to Schrdinger operator. J Math Anal Appl, 2008, 341: 421–432
16 Han Y, Mller D, Yang D C. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 2006, 279: 1505–1537
17 Han Y, Mller D, Yang D C. A theory of Besov and Triebel-Lizorkin spaces on metric measure space modeled on Carnot-Carathéodory spaces. Abstr Appl Anal, 2008, Art. ID 893409, 250 pp
18 Hofmann S, Lu G, Mitrea D, et al. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies- Gaffney estimates. Mem Amer Math Soc, 2011, 1007: 1–78
19 Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344: 37–116
20 Janson S. Mean oscillation and commutators of singular integrals operators. Ark Math, 1978, 16: 263–270
21 Jiang R, Yang D C. Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun Contemp Math, 2011, 13: 331–373
22 Jiang R, Yang D C. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167–1224
23 Jiang R, Yang D C. Generalized vanishing mean oscillation spaces associated with divergence form elliptic operators. Integral Equations Operator Theory, 2010, 67: 123–149
24 Jiang R, Yang D C, Zhou Y. Orlicz-Hardy spaces associated with operators. Sci China Ser A, 2009, 52: 1042–1080
25 Li H. Estimations L p des opéateurs de Schringer sur les groupes nilpotents. J Funct Anal, 1999, 161: 152–218
26 Li P T, Peng L Z. Enpoint estimates for commutators of Riesz transforms associated with Schrdinger operators. Bull Aust Math Soc, 2010, 82: 367–389
27 Liang Y, Yang D C, Yang S. Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates. Sci China Math, 2011, 54: 2395–2426
28 Lin C C, Liu H P, Liu Y. Hardy spaces associated with Schrdinger operators on the Heisenberg group. ArXiv:1106.4960
29 Lin C C, Liu H P. BM O L (H n ) spaces and Carleson measures for Schrdinger operators. Adv Math, 2011, 228: 1631–1688
30 Liu Y, Dong J F. Some estimates of higher order Riesz transform related to Schrdinger type operator. Potential Anal, 2010, 32: 41–55
31 Lu G Z. Embedding theorem into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ Mat, 1996, 40: 301–329
32 Lu S Z, Wu Q, Yang D C. Boundedness of commutators on Hardy type space. Sci China Ser A, 2002, 45: 984-997
33 Macías R, Segovia C. A well-behaved quasi-distance on spaces of homogeneous type. Trab Mat Inst Argentina Mat, 1981, 32: 1–18
34 Nagel A, Stein E M, Wainger S. Balls and metrics defined by vector fields I. Basic properties. Acta Math, 1985, 155: 103–147
35 Pradolini G, Salinas O. Commutators of singular integrals on spaces of homogeneous type. Czechoslov Math J, 2007,57: 75–93
36 Shen Z W. L p -estimates for Schrdinger operators with certain potentials. Ann Inst Fourier Grenoble, 1995, 45: 513–546
37 Sun Y Z, Su W Y. Hardy type estimates for commutators of singular integrals. Sci China Ser A, 2005, 48: 563-575
38 Yan L X. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 2008, 360: 4383–4408
39 Yang D C, Zhou Y. Localized Hardy spaces H1 related to admissible functions on RD-spaces and applications to Schrdinger operators. Trans Amer Math Soc, 2011, 363: 1197–1239
40 Yang D C, Yang D Y, Zhou Y. Localized BMO and BLO spaces on RD-spaces and applications to Schrdinger operators. Commun Pure Appl Anal, 2010, 9: 779–812
41 Yang D C, Yang D Y, Zhou Y. Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrdinger operators. Nagoya Math J, 2010, 198: 77–119
42 Yang D C, Yang S B. Local Hardy spaces of Musielak-Orlicz type and their applications. Sci China Math, 2012, 55: 1677–1720