Abstract
Let L be a one-to-one operator of type ω having a bounded H ∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k ∈ ℕ. In this paper, the authors introduce the Hardy space H p L (ℝn) with p ∈ (0, 1] associated with L in terms of square functions defined via \(\left\{ {e^{ - t^{2k} L} } \right\}_{t > 0}\) and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L 1 with complex bounded measurable coefficients and the 2k-order Schrödinger type operator L 2:= (−Δ)k + V k, where Δ is the Laplacian and 0 ⩽ V ∈ L kloc (ℝn). Moreover, as an application, for i ∈ {1, 2}, the authors prove that the associated Riesz transform ▿ k(L −1/2 i ) is bounded from \(H_{L_i }^p \left( {\mathbb{R}^n } \right)\) to H p(ℝn) for p ∈ (n/(n + k), 1] and establish the Riesz transform characterizations of \(H_{L_1 }^p \left( {\mathbb{R}^n } \right)\) for p ∈ (rn/(n + kr), 1] if \(\left\{ {e^{ - tL_1 } } \right\}_{t > 0}\) satisfies the L r − L 2 k-off-diagonal estimates with r ∈ (1, 2]. These results when k:= 1 and L:= L 1 are known.
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Ahn B, Li J. Orlicz-Hardy spaces associated to oprators satisfying bounded H ∞ functional calculus and Davies-Gaffney estimates. J Math Anal Appl, 2011, 373: 485–501
Albrecht D, Duong X T, McIntosh A. Operator theory and harmonic analysis. In: Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc Centre Math Appl Austral Nat Univ, vol. 34. Canberra: Austral Nat Univ, 1996, 77–136
Auscher P. On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on ℝn and related estimates. Mem Amer Math Soc, 2007, 871: 1–75
Auscher P, Duong X T, McIntosh A. Boundedness of Banach space valued singular integral operators and Hardy spaces. Unpublished preprint, 2005
Auscher P, Hofmann S, Lacey M, et al. The solution of the Kato square root problem for second order elliptic operators on ℝn. Ann of Math (2), 2002, 156: 633–654
Auscher P, Hofmann S, McIntosh A, et al. The Kato square root problem for higher order elliptic operators and systems on ℝn. J Evol Equ, 2001, 1: 361–385
Auscher P, McIntosh A, Russ E. Hardy spaces of differential forms on Riemannian manifolds. J Geom Anal, 2008, 18: 192–248
Auscher P, Russ E. Hardy spaces and divergence operators on strongly Lipschitz domains of ℝn. J Funct Anal, 2003, 201: 148–184
Barbatis G, Davies E. Sharp bounds on heat kernels of higher order uniformly elliptic operators. J Operator Theory, 1996, 36: 179–198
Blunck S, Kunstmann P C. Weak type (p, p) estimates for Riesz transforms. Math Z, 2004, 247: 137–148
Blunck S, Kunstmann P C. Generalized Gaussian estimates and the Legendre transform. J Operator Theory, 2005, 53: 351–365
Cao J, Liu Y, Yang D. Hardy spaces H 1 L (ℝn) associated to Schrödinger type operators (−Δ)2 +V 2. Houston J Math, 2010, 36: 1067–1095
Chang D C, Dafni G, Stein E M. Hardy spaces, BMO and boundary value problems for the Laplacian on a smooth domain in ℝn. Trans Amer Math Soc, 1999, 351: 1605–1661
Chang D C, Krantz S G, Stein E M. Hardy spaces and elliptic boundary value problems. Contemp Math, 1992, 137: 119–131
Chang D C, Krantz S G, Stein E M. H p theory on a smooth domain in ℝN and elliptic boundary value problems. J Funct Anal, 1993, 114: 286–347
Cho Y, Kim J. Atomic decomposition on Hardy-Sobolev spaces. Studia Math, 2006, 177: 25–42
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
Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645
Davies E B. Uniformly elliptic operators with measurable coefficients. J Funct Anal, 1995, 132: 141–169
Duong X T, Li J. Hardy spaces associated to operators satisfying bounded H ∞ functional calculus and Davies-Gaffney estimates. Preprint
Duong X T, Xiao J, Yan L. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13: 87–111
Duong X T, Yan L. New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Comm Pure Appl Math, 2005, 58: 1375–1420
Duong X T, Yan L. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 2005, 18: 943–973
Dziubański J, Zienkiewicz J. Hardy space H 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev Mat Iberoamericana, 1999, 15: 279–296
Dziubański J, Zienkiewicz J. H p spaces for Schrödinger operators. In: Fourier Analysis and Related Topics (Bpolhk edlewo, 2000), Banach Center Publ, vol. 56. Warsaw: Polish Acad Sci, 2002, 45–53
Fefferman C, Stein E M. H p spaces of several variables. Acta Math, 1972, 129: 137–193
García-Cuerva J, Rubio de Francia J. Weighted Norm Inequalities and Related Topics. Amsterdam: North-Holland Publishing Co, 1985
Grafakos L. Classical Fourier Analysis. New York: Springer Press, 2008
Grafakos L, Liu L, Yang D. Boundedness of paraproduct operators on RD-spaces. Sci China Math, 2010, 53: 2097–2114
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
Li B, Bownik M, Yang D, et al. Anisotropic singular integrals in product spaces. Sci China Math, 2010, 53: 3163–3178
Liang Y, Yang D, Yang S. Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates. Sci China Math, 2011, 54: 2395–2426
Haase M. The Functional Calculus for Sectorial Operators. In: Oper Theory Adv Appl, vol. 169. Basel: Birkhäser Verlag, 2006
Han Y, Paluszynski M, Weiss G. A new atomic decomposition for the Triebel-Lizorkin spaces. In: Harmonic Analysis and Operator Theory (Caracas, 1994), Contemp Math, vol. 189. Providence, RI: Amer Math Soc, 1995, 235–249
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
Hofmann S, Martell J. L p bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ Mat, 2003, 47: 497–515
Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344: 37–116
Hofmann S, Mayboroda S. Correction to “Hardy and BMO spaces associated to divergence form elliptic operators”. arXiv:0907.0129
Hofmann S, Mayboroda S, McIntosh A. Second order elliptic operators with complex bounded measurable coefficients in L p, Sobolev and Hardy spaces. Ann Sci École Norm Sup (4), 2011, 44: 723–800
Hu G, Yang D, Zhou Y. Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwanese J Math, 2009, 13: 91–135
Jiang R, Yang D. Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun Contemp Math, 2011, 13: 331–373
Jiang R, Yang D. Predual spaces of Banach completions of Orlicz-Hardy spaces associated with operators. J Fourier Anal Appl, 2011, 17: 1–35
Jiang R, Yang D. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167–1224
Jiang R, Yang D. Generalized vanishing mean oscillation spaces associated with divergence form elliptic operators. Integral Equations Operator Theory, 2010, 67: 123–149
Jiang R, Yang D, Zhou Y. Orlicz-Hardy spaces associated with operators. Sci China Ser A, 2009, 52: 1042–1080
McIntosh A. Operators which have an H ∞ functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), Proc Centre Math Anal Austral Nat Univ, vol. 14. Canberra: Austral Nat Univ, 1986, 210–231
Ouhabaz E M. Analysis of Heat Equations on Domains. In: London Mathematical Society Monographs Series 31. Princeton, NJ: Princeton University Press, 2005
Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993
Stein E M, Weiss G. On the theory of harmonic functions of several variables. I. The theory of H p-spaces. Acta Math, 1960, 103: 25–62
Taibleson M H, Weiss G. The molecular characterization of certain Hardy spaces. In: Astérisque, 77, Representation Theorems for Hardy Spaces. Paris: Soc Math France, 1980, 67–149
Triebel H. Theory of Function Spaces. Basel: Birkhäser Verlag, 1983
Welland G, Zhao S. ɛ-families of operators in Triebel-Lizorkin and tent spaces. Canad J Math, 1995, 47: 1095–1120
Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 2008, 360: 4383–4408
Yang D, Yang D. Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures. Georgian Math J, 2011, 18: 377–397
Yang D C, Yang S B. Local Hardy spaces of Musielak-Orlicz type and their applications. Sci China Math, 2012, 55, doi: 10.1007/s11425-012-4377-z (in press) or arXiv:1108.2797
Yang D, Zhou Y. Localized Hardy spaces H 1 related to admissible functions on RD-spaces and applications to Schrödinger operators. Trans Amer Math Soc, 2011, 363: 1197–1239
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Cao, J., Yang, D. Hardy spaces H p L (ℝn) associated with operators satisfying k-Davies-Gaffney estimates. Sci. China Math. 55, 1403–1440 (2012). https://doi.org/10.1007/s11425-012-4394-y
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DOI: https://doi.org/10.1007/s11425-012-4394-y
Keywords
- Hardy space
- Hardy-Sobolev space
- k-Davies-Gaffney estimate
- Schrödinger type operator
- higher order elliptic operator
- semigroup
- square function
- higher order Riesz transform
- molecule