非齐度量测度空间上的Herz型Hardy空间
|
摘要
设(X, d,μ)是一个同时满足上双倍条件和几何双倍条件的非齐度量测度空间.本文首先引进了非齐度量测度空间上的Herz空间,并利用中心块得到了该空间的分解定理.然后,根据离散系数K_(B,S)~((ρ),p),引入了非齐度量测度空间上的原子Herz型Hardy空间与分子Herz型Hardy空间,并证明了原子Herz型Hardy空间和分子Herz型Hardy空间的等价性.最后作为应用,本文讨论了Calderón-Zygmund算子在这些空间上的有界性.
Let(X, d, μ) be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling conditions. In this paper, the Herz spaces on the non-homogeneous metric measure space are introduced. Then the decomposition of the Herz space by the central blocks is obtained. The atomic Herz type Hardy spaces and the molecular Herz type Hardy spaces on the non-homogeneous metric measure space via the discrete coefficient K_(B,S)~((ρ),p),are also defined. In addition, the equivalence of the atomic Herz type Hardy spaces and the molecular Herz type Hardy spaces is established. As the applications of these spaces, some boundedness of Calderón-Zygmund operators on the Herz type Hardy spaces are discussed.
Let(X, d, μ) be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling conditions. In this paper, the Herz spaces on the non-homogeneous metric measure space are introduced. Then the decomposition of the Herz space by the central blocks is obtained. The atomic Herz type Hardy spaces and the molecular Herz type Hardy spaces on the non-homogeneous metric measure space via the discrete coefficient K_(B,S)~((ρ),p),are also defined. In addition, the equivalence of the atomic Herz type Hardy spaces and the molecular Herz type Hardy spaces is established. As the applications of these spaces, some boundedness of Calderón-Zygmund operators on the Herz type Hardy spaces are discussed.
引文
1 Fefferman C,Stein E M.Hpspaces of several variables.Acta Math,1972,129:137-193
2 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
3 Cao J,Yang D.Hardy spaces HLp(Rn)associated with operators satisfying k-Davies-Gaffney estimates.Sci China Math,2012,55:1403-1440
4 Fu X,Yang D.Products of functions in H1ρ(X)and BMOρ(X)over RD-spaces and applications to Schr¨odinger operators.J Geom Anal,2017,27:2938-2976
5 Jiang R J,Yang D,Zhou Y.Orlicz-Hardy spaces associated with operators.Sci China Ser A,2009,52:1042-1080
6陈杰诚,王慧.奇异积分算子在乘积Tirebel-Lizorkin空间.中国科学A辑,2009,39:861-872
7 Li B D,Bownik M,Yang D,et al.Anisotropic singular integrals in product spaces.Sci China Math,2010,53:3163-3178
8 Liu J,Yang D,Yuan W.Anisotropic Hardy-Lorentz spaces and their applications.Sci China Math,2016,59:1669-1720
9刘丽光,杨大春.RD-空间上Hardy空间的极大函数特征及其应用.中国科学A辑,2008,38:490-518
10 Yang D,Yang S.Local Hardy spaces of Musielak-Orlicz type and their applications.Sci China Math,2012,55:1677-1720
11 Zhao K.Hardy-Hausdorff spaces on the Heisenberg group.Sci China Math,2016,59:2167-2184
12 Fan D,Yang D.Herz-type Hardy spaces on Vilenkin groups and their applications.Sci China Ser A,2000,43:481-494
13 Lu S.Herz type spaces.Adv Math(China),2004,33:257-272
14 Lu S,Tang L,Yang D.Boundedness of commutators on homogeneous Herz spaces.Sci China Ser A,1998,41:1023-1033
15陆善镇,杨大春.加权Herz型的Hardy空间及其应用.中国科学A辑,1995,25:235-245
16 Lu S,Yang D.Some characterizations of the Hardy spaces associated with a kind of block space.J Beijing Normal Univ(Natur Sci),1991,27:269-279
17 Lu S,Yang D.Herz-type Sobolev and Bessel potential spaces and their applications.Sci China Ser A,1997,40:113-129
18 Lu S,Yang D.The decomposition of weighted Herz space on Rnand their applications.Sci China Math,1995,38:147-158
19 Lu S,Yang D,Hu G.Herz Type Spaces and Their Applications.Beijing:Science Press,2008
20 Grafakos L,Liu L,Yang D.Boundedness of paraproduct operators on RD-spaces.Sci China Math,2010,53:2097-2114
21 Han Y,Yang D.Triebel-Lizorkin spaces with non-doubling measures.Studia Math,2004,162:105-140
22 Hu G,Meng Y,Yang D.Endpoint estimate for maximal commutators with non-doubling measues.Acta Math Sci,2006,26:271-280
23 Hu G,Yang D,Yang D.Weighted estimates for commutators of singular integral operators with non-doubling measures.Sci China Ser A,2007,50:1621-1641
24 Lin H,Meng Y,Yang D.Weighted estimates for commutators of multilinear Calder′on-Zygmund operators with nondoubling measures.Acta Math Sci,2010,30:1-18
25 Tolsa X.BMO,H1,and Calder′on-Zygmund operators for non doubling measures.Math Ann,2001,319:89-149
26 Tolsa X.Littlewood-Paley theory and the T(1)theorem with non-doubling measures.Adv Math,2001,164:57-116
27 Yang D,Yang D,Hu G.The Hardy Space H1with Non-Doubling Measures and Their Applications.Lecture Notes in Mathematics,vol.2084.Berlin:Springer-Verlag,2013
28 Hyt¨onen T.A framework for non-homogeneous analysis on metric spaces,and the RBMO space of Tolsa.Publ Mat,2010,54:485-504
29 Fu X,Yang D,Yuan W.Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces.Taiwanese J Math,2014,18:509-557
30 Hu G,Meng Y,Yang D.Weighted norm inequalities for multilinear Calder′on-Zygmund operators on non-homogeneous metric measure spaces.Forum Math,2014,26:1289-1322
31 Lin H,Wu S,Yang D.Boundedness of certain commutators over non-homogeneous metric measure spaces.Anal Math Phys,2017,7:187-218
32 Lin H,Yang D.Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces.Sci China Math,2014,57:123-144
33 Lin H,Yang D.An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces.Banach J Math Anal,2012,6:168-179
34 Lin H,Yang D.Spaces of type BLO on non-homogeneous metric measure.Front Math China,2011,6:271-292
35 Liu S,Yang D,Yang D.Boundedness of Calder′on-Zygmund operators on non-homogeneous metric measure spaces:Equivalent characterizations.J Math Anal Appl,2012,386:258-272
36 Liu S,Meng Y,Yang D.Boundedness of maximal Calder′on-Zygmund operators on non-homogeneous metric measure spaces.Proc Roy Soc Edinburgh Sect A,2014,144:567-589
37 Fu X,Lin H B,Yang D,et al.Hardy spaces Hpover non-homogeneous metric measure spaces and their applications.Sci China Math,2015,58:309-388
38 Hyt¨onen T,Yang D,Yang D.The Hardy space H1on non-homogeneous metric spaces.Math Proc Cambridge Philos Soc,2012,153:9-31
39 Coifman R R,Weiss G.Analyse Harmonique Non-Commutative sur Certains Espaces Homog`enes.Lecture Notes in Mathematics,vol.242.Berlin-New York:Springer-Verlag,1971
40 Heinonen J.Lectures on Analysis on Metric Spaces.New York:Springer-Verlag,2001
41 Tan C,Li J.Littlewood-Paley theory on metric spaces with non doubling measures and its applications.Sci China Math,2015,58:983-1004
42 Gong R,Li J,Yan L.A local version of Hardy spaces associated with operators on metric spaces.Sci China Math,2013,56:315-330
43 Hytonen T,Martikainen H.Non-homogeneous T b theorem and random dyadic cubes on metric measure spaces.JGeom Anal,2012,22:1071-1107
44 Bui T A,Duong X T.Hardy spaces,regularized BMO spaces and the boundedness of Calder′on-Zygmund operators on non-homogeneous spaces.J Geom Anal,2013,23:895-932
45 Hytonen T,Liu S,Yang D,et al.Boundedness of Calder′on-Zygmund operators on non-homogeneous metric measure spaces.Canad J Math,2012,64:892-923
2 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
3 Cao J,Yang D.Hardy spaces HLp(Rn)associated with operators satisfying k-Davies-Gaffney estimates.Sci China Math,2012,55:1403-1440
4 Fu X,Yang D.Products of functions in H1ρ(X)and BMOρ(X)over RD-spaces and applications to Schr¨odinger operators.J Geom Anal,2017,27:2938-2976
5 Jiang R J,Yang D,Zhou Y.Orlicz-Hardy spaces associated with operators.Sci China Ser A,2009,52:1042-1080
6陈杰诚,王慧.奇异积分算子在乘积Tirebel-Lizorkin空间.中国科学A辑,2009,39:861-872
7 Li B D,Bownik M,Yang D,et al.Anisotropic singular integrals in product spaces.Sci China Math,2010,53:3163-3178
8 Liu J,Yang D,Yuan W.Anisotropic Hardy-Lorentz spaces and their applications.Sci China Math,2016,59:1669-1720
9刘丽光,杨大春.RD-空间上Hardy空间的极大函数特征及其应用.中国科学A辑,2008,38:490-518
10 Yang D,Yang S.Local Hardy spaces of Musielak-Orlicz type and their applications.Sci China Math,2012,55:1677-1720
11 Zhao K.Hardy-Hausdorff spaces on the Heisenberg group.Sci China Math,2016,59:2167-2184
12 Fan D,Yang D.Herz-type Hardy spaces on Vilenkin groups and their applications.Sci China Ser A,2000,43:481-494
13 Lu S.Herz type spaces.Adv Math(China),2004,33:257-272
14 Lu S,Tang L,Yang D.Boundedness of commutators on homogeneous Herz spaces.Sci China Ser A,1998,41:1023-1033
15陆善镇,杨大春.加权Herz型的Hardy空间及其应用.中国科学A辑,1995,25:235-245
16 Lu S,Yang D.Some characterizations of the Hardy spaces associated with a kind of block space.J Beijing Normal Univ(Natur Sci),1991,27:269-279
17 Lu S,Yang D.Herz-type Sobolev and Bessel potential spaces and their applications.Sci China Ser A,1997,40:113-129
18 Lu S,Yang D.The decomposition of weighted Herz space on Rnand their applications.Sci China Math,1995,38:147-158
19 Lu S,Yang D,Hu G.Herz Type Spaces and Their Applications.Beijing:Science Press,2008
20 Grafakos L,Liu L,Yang D.Boundedness of paraproduct operators on RD-spaces.Sci China Math,2010,53:2097-2114
21 Han Y,Yang D.Triebel-Lizorkin spaces with non-doubling measures.Studia Math,2004,162:105-140
22 Hu G,Meng Y,Yang D.Endpoint estimate for maximal commutators with non-doubling measues.Acta Math Sci,2006,26:271-280
23 Hu G,Yang D,Yang D.Weighted estimates for commutators of singular integral operators with non-doubling measures.Sci China Ser A,2007,50:1621-1641
24 Lin H,Meng Y,Yang D.Weighted estimates for commutators of multilinear Calder′on-Zygmund operators with nondoubling measures.Acta Math Sci,2010,30:1-18
25 Tolsa X.BMO,H1,and Calder′on-Zygmund operators for non doubling measures.Math Ann,2001,319:89-149
26 Tolsa X.Littlewood-Paley theory and the T(1)theorem with non-doubling measures.Adv Math,2001,164:57-116
27 Yang D,Yang D,Hu G.The Hardy Space H1with Non-Doubling Measures and Their Applications.Lecture Notes in Mathematics,vol.2084.Berlin:Springer-Verlag,2013
28 Hyt¨onen T.A framework for non-homogeneous analysis on metric spaces,and the RBMO space of Tolsa.Publ Mat,2010,54:485-504
29 Fu X,Yang D,Yuan W.Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces.Taiwanese J Math,2014,18:509-557
30 Hu G,Meng Y,Yang D.Weighted norm inequalities for multilinear Calder′on-Zygmund operators on non-homogeneous metric measure spaces.Forum Math,2014,26:1289-1322
31 Lin H,Wu S,Yang D.Boundedness of certain commutators over non-homogeneous metric measure spaces.Anal Math Phys,2017,7:187-218
32 Lin H,Yang D.Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces.Sci China Math,2014,57:123-144
33 Lin H,Yang D.An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces.Banach J Math Anal,2012,6:168-179
34 Lin H,Yang D.Spaces of type BLO on non-homogeneous metric measure.Front Math China,2011,6:271-292
35 Liu S,Yang D,Yang D.Boundedness of Calder′on-Zygmund operators on non-homogeneous metric measure spaces:Equivalent characterizations.J Math Anal Appl,2012,386:258-272
36 Liu S,Meng Y,Yang D.Boundedness of maximal Calder′on-Zygmund operators on non-homogeneous metric measure spaces.Proc Roy Soc Edinburgh Sect A,2014,144:567-589
37 Fu X,Lin H B,Yang D,et al.Hardy spaces Hpover non-homogeneous metric measure spaces and their applications.Sci China Math,2015,58:309-388
38 Hyt¨onen T,Yang D,Yang D.The Hardy space H1on non-homogeneous metric spaces.Math Proc Cambridge Philos Soc,2012,153:9-31
39 Coifman R R,Weiss G.Analyse Harmonique Non-Commutative sur Certains Espaces Homog`enes.Lecture Notes in Mathematics,vol.242.Berlin-New York:Springer-Verlag,1971
40 Heinonen J.Lectures on Analysis on Metric Spaces.New York:Springer-Verlag,2001
41 Tan C,Li J.Littlewood-Paley theory on metric spaces with non doubling measures and its applications.Sci China Math,2015,58:983-1004
42 Gong R,Li J,Yan L.A local version of Hardy spaces associated with operators on metric spaces.Sci China Math,2013,56:315-330
43 Hytonen T,Martikainen H.Non-homogeneous T b theorem and random dyadic cubes on metric measure spaces.JGeom Anal,2012,22:1071-1107
44 Bui T A,Duong X T.Hardy spaces,regularized BMO spaces and the boundedness of Calder′on-Zygmund operators on non-homogeneous spaces.J Geom Anal,2013,23:895-932
45 Hytonen T,Liu S,Yang D,et al.Boundedness of Calder′on-Zygmund operators on non-homogeneous metric measure spaces.Canad J Math,2012,64:892-923