Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ;L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(ℝn) and the classical Musielak-Orlicz-Hardy space Hv(ℝn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(ℝn) associated with the second order elliptic operator in divergence form on ℝn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(ℝn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak-Orlicz space Lφ(ℝn) when i(φ) ∊ (0; 1], from Hφ,L(ℝn) to Hφ(ℝn) when i(φ) ∊ ( [...] ; 1], and from Hφ,L(ℝn) to the weak Musielak-Orlicz-Hardy space WHφ(ℝn) when i(φ)= [...] is attainable and φ(·; t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ
@article{bwmeta1.element.doi-10_2478_agms-2012-0006,
author = {The Anh Bui and Jun Cao and Luong Dang Ky and Dachun Yang and Sibei Yang},
title = {Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates},
journal = {Analysis and Geometry in Metric Spaces},
volume = {1},
year = {2013},
pages = {69-129},
zbl = {1261.42034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2012-0006}
}
The Anh Bui; Jun Cao; Luong Dang Ky; Dachun Yang; Sibei Yang. Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates. Analysis and Geometry in Metric Spaces, Tome 1 (2013) pp. 69-129. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2012-0006/
D. Albrecht, X. T. Duong and A. McIntosh, Operator theory and harmonic analysis, Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), 77-136, Proc. Centre Math. Appl. Austral. Nat. Univ., 34, Austral. Nat. Univ., Canberra, 1996. | Zbl 0903.47010
T. Aoki, Locally bounded linear topological space, Proc. Imp. Acad. Tokyo 18 (1942), 588-594.[Crossref] | Zbl 0060.26503
J. Assaad and E. M. Ouhabaz, Riesz transforms of Schrödinger operators on manifolds, J. Geom. Anal. 22 (2012), 1108-1136. | Zbl 1259.58006
P. Auscher, On necessary and sufficient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on Rn and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75 pp. | Zbl 1221.42022
P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005.
P. Auscher and J. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. general operator theory and weights, Adv. Math. 212 (2007), 225-276. | Zbl 1213.42030
P. Auscher and J. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), 265-316.[Crossref] | Zbl 1210.42023
P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192-248.[Crossref] | Zbl 1217.42043
P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172 pp. | Zbl 0909.35001
S. Blunck and P. C. Kunstmann, Weak type (p; p) estimates for Riesz transforms, Math. Z. 247 (2004), 137-148. | Zbl 1138.35315
S. Blunck and P. Kunstmann, Generalized Gaussian estimates and the Legendre transform, J. Operator Theory 53 (2005), 351-365. | Zbl 1117.47020
T. A. Bui, J. Cao, L. Ky, D. Yang and S. Yang, Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Taiwanese J. Math. (to appear). | Zbl 1284.42066
A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat. 54 (2010), 341-358.[Crossref] | Zbl 1205.42020
A. Bonami and S. Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces, Colloq. Math. 118 (2010), 107-132. | Zbl 1195.32002
A. Bonami, S. Grellier and L. D. Ky, Paraproducts and products of functions in BMO(Rn) and H1(Rn) through wavelets, J. Math. Pure Appl. 97 (2012), 230-241. | Zbl 1241.47028
A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in BMO and H1, Ann. Inst. Fourier (Grenoble) 57 (2007), 1405-1439. | Zbl 1132.42010
T. A. Bui and X. T. Duong, Weighted Hardy spaces associated to operators and boundedness of singular integrals, arXiv: 1202.2063. | Zbl 1301.42023
J. Cao and D. Yang, Hardy spaces HpL (Rn) associated to operators satisfying k-Davies-Gaffney estimates, Sci. China Math. 55 (2012), 1403-1440.[Crossref] | Zbl 1266.42057
J. Cao, D. Yang and S. Yang, Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators, Rev. Mat. Complut. 26 (2013), 99-114. | Zbl 1298.47029
R. R. Coifman, A real variable characterization of Hp, Studia Math. 51 (1974), 269-274. | Zbl 0289.46037
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. | Zbl 0864.42009
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.[Crossref] | Zbl 0569.42016
R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homog`enes, Lecture Notes in Math., 242, Springer, Berlin, 1971.
T. Coulhon and A. Sikora, Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem, Proc. Lond. Math. Soc. 96 (2008), 507-544. | Zbl 1148.35009
M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded H1 functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51-89. | Zbl 0853.47010
D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), 2941-2960. | Zbl 0851.42016
E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141-169. | Zbl 0839.35034
L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), 657-700. | Zbl 1096.46013
L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), 1731-1768. | Zbl 1179.46028
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.[Crossref] | Zbl 1078.42013
C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-195. | Zbl 0257.46078
M. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 1-11.[Crossref] | Zbl 0102.09202
J. García-Cuerva, Weighted Hp spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 1-63.
J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North- Holland, 1985.
L. Grafakos, Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics 250, Springer, New York, 2009. | Zbl 1158.42001
L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 17-35. | Zbl 0848.58051
E. Harboure, O. Salinas and B. Viviani, A look at BMO(!) through Carleson measures, J. Fourier Anal. Appl. 13 (2007), 267-284.[Crossref] | Zbl 1174.42019
M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäser Verlag, Basel, 2006.
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. | Zbl 1232.42018
S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37-116. | Zbl 1162.42012
S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in Lp, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4) 44 (2011), 723-800. | Zbl 1243.47072
S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, arXiv: 1201.1945. | Zbl 1285.42020
T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143-161. | Zbl 0802.35016
S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959-982. | Zbl 0453.46027
R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math. 13 (2011), 331-373.[Crossref] | Zbl 1221.42042
R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 1167-1224. | Zbl 1205.46014
R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators, Sci. China Ser. A 52 (2009), 1042-1080.[Crossref] | Zbl 1177.42018
R. Johnson and C. J. Neugebauer, Homeomorphisms preserving Ap, Rev. Mat. Ibero. 3 (1987), 249-273. | Zbl 0677.42019
L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, arXiv: 1103.3757. | Zbl 1284.42073
L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. (2012), DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05727-8.[Crossref]
L. D. Ky, Endpoint estimates for commutators of singular integrals related to Schrödinger operators, arXiv:1203.6335. | Zbl 06539486
L. D. Ky, Bilinear decompositions for the product space H1 L×BMOL, arXiv:1204.3041.
L. D. Ky, On weak -convergence in H1 L (Rd), arXiv:1205.2542.
R. H. Latter, A characterization of Hp(Rn) in terms of atoms, Studia Math. 62 (1978), 93-101. | Zbl 0398.42017
A. K. Lerner, Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces, Math. Z. 251 (2005), 509-521. | Zbl 1092.42009
Y. Liang, J. Huang and D. Yang, New real-variable characterizations of Hardy spaces of Musielak-Orlicz type, J. Math. Anal. Appl. 395 (2012), 413-428. | Zbl 1256.42035
Y. Liang, D. Yang and S. Yang, Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates, Sci. China Math. 54 (2011), 2395-2426.[Crossref] | Zbl 1245.42019
A. McIntosh, Operators which have an H1 functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986), 210-231, Proc. Centre Math. Anal., Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, 1986.
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983. | Zbl 0557.46020
E. Nakai, Pointwise multipliers on weighted BMO spaces, Studia Math. 125 (1997), 35, 35-56. | Zbl 0874.42009
E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan 37 (1985), 207-218. | Zbl 0546.42019
M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991. | Zbl 0724.46032
M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 250, Marcel Dekker, Inc., New York, 2002. | Zbl 0997.46027
S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 471-473. | Zbl 0079.12602
E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, 125-135, Proc. Centre Math. Appl., 42, Austral. Nat. Univ., Canberra, 2007.
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. | Zbl 0836.35030
L. Song and L. Yan, Riesz transforms associated to Schrödinger operators on weighted Hardy spaces, J. Funct. Anal. 259 (2010), 1466-1490. | Zbl 1202.35072
J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511-544. | Zbl 0429.46016
J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989. | Zbl 0676.42021
E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of Hp-spaces, Acta Math. 103 (1960), 25-62. | Zbl 0097.28501
D. Yang and S. Yang, Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of Rn, Indiana Univ. Math. J. (to appear) or arXiv:1107.2971. | Zbl 1271.42034
D. Yang and S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of Rn, Rev. Mat. Ibero. âA.(2013), DOI 10.4171/RMI/719.[Crossref] | Zbl 1266.42056
D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math. 55 (2012), 1677-1720.[Crossref] | Zbl 1266.42055
D. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators and their applications, J. Geom. Anal. (2012), DOI: 10.1007/s12220-012-9344-y or arXiv: 1201.5512.[Crossref] | Zbl 1302.42033
K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995. | Zbl 0842.92020