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Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry

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Abstract

We prove that if \(\tau \) is a large positive number, then the atomic Goldberg-type space \({\mathfrak {h}}^1(N)\) and the space \({\mathfrak {h}}_{{\mathscr {R}}_\tau }^1(N)\) of all integrable functions on N of which local Riesz transform \({\mathscr {R}}_\tau \) is integrable, are the same space on any complete noncompact Riemannian manifold N with Ricci curvature bounded from below and positive injectivity radius. We also relate \({\mathfrak {h}}^1(N)\) to a space of harmonic functions on the slice \(N\times (0,\delta )\) for \(\delta >0\) small enough.

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References

  1. Alvarado, R., Mitrea, M.: Hardy Spaces on Ahlfors-Regular Quasi Metric Space. A sharp theory. Lecture Notes in Mathematics, vol. 2142. Springer, New York (2015)

    MATH  Google Scholar 

  2. Anker, J.-Ph.: Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65, 257–297 (1992)

    MathSciNet  MATH  Google Scholar 

  3. Anker, J.-Ph., Dziubański, J., Heina, A.: Harmonic functions, conjugate harmonic functions and the hardy space \(H^1\) in the rational Dunkl setting. J. Fourier Anal. Appl. 25, 2356s–22418 (2019)

    MathSciNet  Google Scholar 

  4. Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18, 192–248 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Bakry, D.: Études des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. Sém. Prob. Strasbourg XX I, 137–172 (1987)

    MATH  Google Scholar 

  6. Carbonaro, A., Mauceri, G., Meda, S.: \(H^1\), \(BMO\) and singular integrals for certain metric measure spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. 8, 543–582 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Carbonaro, A., Mauceri, G., Meda, S.: \(H^1\) and \(BMO\) for certain locally doubling metric measure spaces of finite measure. Colloq. Math. 118, 13–41 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Chavel, I., Feldman, E.A.: Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds. Duke Math. J. 64, 473–499 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Cheeger, J., Colding, T.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144, 189–237 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Christ, M., Geller, D.: Singular integral characterizations of Hardy spaces on homogeneous groups. Duke Math. J. 51, 547–598 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Coifman, R.R.: A real variable characterisation of \(H^p\). Studia Math. 51, 269–274 (1974)

    MathSciNet  MATH  Google Scholar 

  13. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Cowling, M.G.: Harmonic analysis on semigroups. Ann. Math. 117, 267–283 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Cowling, M., Meda, S.: Harmonic analysis and ultracontractivity. Trans. Am. Math. Soc. 340(2), 733–752 (1993)

    MathSciNet  MATH  Google Scholar 

  16. Croke, C.B.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. École Norm. Sup. (4) 13(4), 419–435 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  18. Davies, E.B.: Pointwise bounds on the space and time derivatives of heat kernels. J. Oper. Theory 1, 367–378 (1989)

    MathSciNet  MATH  Google Scholar 

  19. Dindoš, M.: Hardy spaces and potential theory on \(C^1\) domains in Riemannian manifolds. Mem. Am. Math. Soc. 191(894), vi+78 (2008)

    MATH  Google Scholar 

  20. Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory, Wiley Classic Library Edition, (1988)

  21. Dziubański, J., Jotsaroop, K.: On hardy and BMO spaces for Grushin operator. J. Fourier Anal. Appl. 22, 954–995 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Dziubański, J., Zienkiewicz, J.: Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15, 279–296 (1999)

    MATH  Google Scholar 

  23. Dziubański, J., Zienkiewicz, J.: A characterization of Hardy spaces associated with certain Schrödinger operators. Potential Anal. 41, 917–930 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Erdélyi, A.: Asymptotic expansions. Dover Publications Inc, New York (1956)

    MATH  Google Scholar 

  25. Fabes, E., Kenig, C.: On Hardy space \(H^1\) of a \(C^1\) domain. Arkiv för Matematik 19, 1–22 (1981)

    MathSciNet  MATH  Google Scholar 

  26. Fefferman, C., Stein, E.M.: Hardy spaces of several variables. Acta Math. 129, 137–193 (1972)

    MathSciNet  MATH  Google Scholar 

  27. Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn., p. xii+324. Princeton University Press, Princeton, NJ (1995)

    MATH  Google Scholar 

  28. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Mathematical Notes, Vol. 28. Princeton University Press, Princeton (1982). xii+285 pp

  29. Gauthier, P.H., Grothmann, R., Hengartner, W.: Asymptotic maximum principles for subharmonic and plurisubharmonic functions. Can. J. Math. XL(2), 477–486 (1988)

    MathSciNet  MATH  Google Scholar 

  30. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Classics in Mathematics, Springer, Berlin (2001)

    MATH  Google Scholar 

  31. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)

    MathSciNet  MATH  Google Scholar 

  32. Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, III Springer, Berlin (2014)

    Google Scholar 

  33. Grafakos, L., Liu, L., Yang, D.: Radial maximal function characterization for Hardy spaces on RD-spaces Bull. Soc. Math. Fr. 137, 225–251 (2009)

    MATH  Google Scholar 

  34. Grigor’yan, A.: Heat kernel and analysis on manifolds, Studies in Advanced Mathematics, Vol. 47, American Mathematical Society, International Press (2009)

  35. Haase, M.: The functional calculus for sectorial operators, Operator theory, Advances and applications 169, Birkhäuser Verlag (2006)

  36. Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces. Abstr. Appl. Anal., art. ID 893409, 250 pp (2008). https://doi.org/10.1155/2008/893409

  37. Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics 5, Amer. Math. Soc. (1999)

  38. Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney Estimates. Mem. Amer. Math. Soc. 214(1007) (2011)

  39. Ionescu, A.D.: Fourier integral operators on noncompact symmetric spaces of real rank one. J. Funct. Anal. 174, 274–300 (2000)

    MathSciNet  MATH  Google Scholar 

  40. Latter, R.: A decomposition of \(H^p(\mathbb{R}^n)\) in terms of atoms. Studia Math. 62, 92–101 (1978)

    MathSciNet  Google Scholar 

  41. Lebedev, N.N.: Special Functions and Their Applications. Dover Publications, Mineola (1972)

    MATH  Google Scholar 

  42. Lohoué, N.: Transformées de Riesz et fonctions sommables. Am. J. Math. 114, 875–922 (1992)

    MATH  Google Scholar 

  43. Maheux, P., Saloff-Coste, L.: Analyse sur les boules d’un opérateur sous-elliptique. Math. Ann. 303(4), 713–740 (1995)

    MathSciNet  MATH  Google Scholar 

  44. Martini, A., Meda, S., Vallarino, M.: A family of Hardy type spaces on nondoubling manifolds. Ann. Mat. Pura Appl. 199, 2061–2085 (2020). https://doi.org/10.1007/s10231-020-00956-9

    Article  MathSciNet  MATH  Google Scholar 

  45. Martini, A., Meda, S., Vallarino, M.: Maximal characterization of local Hardy spaces on locally doubling manifolds. Math. Z. (2021). https://doi.org/10.1007/s00209-021-02856-x

  46. Martini, A., Meda, S., Vallarino, M., Veronelli, G.: Comparison of Hardy type spaces defined via maximal functions and the Riesz transform on certain nondoubling manifolds, preprint

  47. Mauceri, G., Meda, S., Vallarino, M.: Weak type \(1\) estimates for functions of the Laplace-Beltrami operator on manifolds with bounded geometry. Math. Res. Lett. 16, 861–879 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Mauceri, G., Meda, S., Vallarino, M.: Hardy-type spaces on certain noncompact manifolds and applications. J. Lond. Math. Soc. 84, 243–268 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Mauceri, G., Meda, S., Vallarino, M.: Atomic decomposition of Hardy type spaces on certain noncompact manifolds. J. Geom. Anal. 22, 864–891 (2012)

    MathSciNet  MATH  Google Scholar 

  50. Mauceri, G., Picardello, M., Ricci, F.: A Hardy space associated with twisted convolution. Adv. Math. 39, 270–288 (1981)

    MathSciNet  MATH  Google Scholar 

  51. Meda, S., Volpi, S.: Spaces of Goldberg type on certain measured metric spaces. Ann. Mat. Pura Appl. 196, 947–981 (2017)

    MathSciNet  MATH  Google Scholar 

  52. Oberhettinger, F.: Tables of Fourier Transforms and Fourier Transforms of Distributions, Pure and Applied Mathematics, vol. 289. Springer, New York (1990)

    MATH  Google Scholar 

  53. Russ, E.: \(H^1\)-\(L^1\) boundedness of Riesz transforms on Riemannian manifolds and on graphs. Pot. Anal. 14, 301–330 (2001)

    MathSciNet  MATH  Google Scholar 

  54. Saloff-Coste, L.: Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  55. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton N. J. (1970)

    MATH  Google Scholar 

  56. Stein, E.M.: Harmonic Analysis. Real variable methods, orthogonality and oscillatory integrals, Princeton Math. Series No. 43, Princeton N. J. (1993)

  57. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood–Paley Theory. Annals of Math. Studies, No. 63, Princeton N. J. (1970)

  58. Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables. I. The theory of \(H^p\)-spaces. Acta Math. 103, 25–62 (1960)

    MathSciNet  MATH  Google Scholar 

  59. Strichartz, R.S.: The Hardy space \(H^1\) on manifolds and submanifolds. Can. J. Math. 24, 915–925 (1972)

    MATH  Google Scholar 

  60. Taylor, M.E.: Hardy spaces and bmo on manifolds with bounded geometry. J. Geom. Anal. 19, 137–190 (2009)

    MathSciNet  MATH  Google Scholar 

  61. Taylor, M.E.: \(L^p\) estimates on functions of the Laplace operator. Duke Math. J. 58, 773–793 (1989)

    MathSciNet  MATH  Google Scholar 

  62. Tolsa, X.: \(BMO\), \(H^1\), and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319, 89–149 (2001)

    MathSciNet  MATH  Google Scholar 

  63. Uchiyama, A.: A constructive proof of the Fefferman-Stein decomposition of \(BMO(\mathbb{R}^n)\). Acta Math. 148, 215–241 (1982)

    MathSciNet  MATH  Google Scholar 

  64. Volpi, S.: Bochner-Riesz means of eigenfunction expansions and local Hardy spaces on manifolds with bounded geoemetry, Ph.D Thesis, Università di Milano–Bicocca, Milano (2012)

  65. Yang, D., Zhou, Y.: Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications. Math. Ann. 346, 307–333 (2010)

    MathSciNet  MATH  Google Scholar 

  66. Yang, D., Zhou, Y.: Localized Hardy spaces \(H^1\) related to admissible functions on RD-spaces and applications to Schrödinger operators. Trans. Am. Math. Soc. 363, 1197–1239 (2011)

    MATH  Google Scholar 

  67. Zhang, Qi. S.: The Boundary Behavior of Heat Kernels of Dirichlet Laplacians. J. Differ. Equ. 182, 416–430 (2002)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Alessio Martini, Alberto Setti and Maria Vallarino for valuable conversations on the subject of this paper.

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  1. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via R. Cozzi 55, 20125, Milano, Italy

    Stefano Meda & Giona Veronelli

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  1. Stefano Meda
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  2. Giona Veronelli
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Meda, S., Veronelli, G. Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry. J Geom Anal 32, 55 (2022). https://doi.org/10.1007/s12220-021-00810-1

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