Advances in Operator Theory

Atomic characterizations of Hardy spaces associated to Schrödinger type operators

Junqiang Zhang and Zongguang Liu

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Abstract

‎In this article‎, ‎the authors consider the Schrödinger type‎ ‎operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$‎, ‎where the matrix $A$ is symmetric and satisfies‎ ‎the uniformly elliptic condition and the nonnegative potential‎ ‎$V$ belongs to the reverse Hölder class $RH_q(\mathbb{R}^n)$‎ ‎with $q\in(n/2,\,\infty)$‎. ‎Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function‎ ‎satisfying the globally $\log$-Hölder continuous condition‎. ‎The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$‎ ‎and establish its atomic characterization‎. ‎The atoms here are closer to the atoms of‎ ‎variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$ in spirit‎, ‎which further implies that $H^{p(\cdot)}(\mathbb{R}^n)$ is continuously embedded in‎ ‎$H_L^{p(\cdot)}(\mathbb{R}^n)$‎.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 3 (2019), 604-624.

Dates
Received: 27 November 2018
Accepted: 19 December 2018
First available in Project Euclid: 2 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.aot/1551495623

Digital Object Identifier
doi:10.15352/aot.1811-1440

Mathematical Reviews number (MathSciNet)
MR3919034

Zentralblatt MATH identifier
07056788

Subjects
Primary: 42B30: $H^p$-spaces
Secondary: 42B35‎ ‎35J10

Keywords
Hardy space‎ Schrödinger type operator ‎‎variable exponent ‎ ‎atom‎

Citation

Zhang, Junqiang; Liu, Zongguang. Atomic characterizations of Hardy spaces associated to Schrödinger type operators. Adv. Oper. Theory 4 (2019), no. 3, 604--624. doi:10.15352/aot.1811-1440. https://projecteuclid.org/euclid.aot/1551495623


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References

  • D. Albrecht, X. T. Duong, and A. McIntosh, Operator theory and harmonic analysis, in: 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.
  • 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Auscher, X. T. Duong, and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished manuscript, 2005.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 E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $\rn$, J. Funct. Anal. 201 (2003), no. 1, 148–184.
  • P. Auscher and P. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque No. 249 (1998), viii+172 pp.
  • J. Cao, D.-C. Chang, D. Yang, and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal. 13 (2014), no. 4, 1435–1463.
  • R. R. Coifman, A real variable characterization of $H^p$, Studia Math. 51 (1974), 269–274.
  • R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335.
  • D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.
  • D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264.
  • D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  • L. Diening, P. Harjulehto, P. Hästö, and M. R$\mathring{\rm u}$žička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011.
  • X. T. Duong and L. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), no. 10, 1375–1420.
  • 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), no. 4, 943–973.
  • J. Dziubański and J. Zienkiewicz, Hardy space $H^1$ associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoamericana 15 (1999), no. 2, 279–296.
  • J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators, in Fourier analysis and related topics, Banach Center Publ., 56, Polish Acad. Sci. Inst. Math., Warsaw, (2002), 45–53.
  • J. Dziubański and J. Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math. 98 (2003), no. 1, 5–38.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129, 1972, no. 3-4, 137–193.
  • 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.
  • S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37–116.
  • R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), no. 4, 1167–1224.
  • T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
  • R. H. Latter, A decomposition of $H^p(\rn)$ in terms of atoms, Studia Math., 62 (1978), no. 1, 93–101.
  • E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  • Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148.
  • Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546.
  • E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces, Acta Math. 103 (1960), 25–62.
  • L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4383–4408.
  • S. Yang, Some estimates for Schrödinger type operators on Musielak-Orlicz-Hardy spaces, Taiwanese J. Math. 18 (2014), no. 4, 1293-1328.
  • D. Yang and J. Zhang, Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates on metric measure spaces of homogeneous type, Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 1, 47–87.
  • D. Yang, J. Zhang, and C. Zhuo, Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Proc. Edinb. Math. Soc. 61 (2018), no. 3, 759–810.
  • C. Zhuo, D. Yang, and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1541–1577.