Revista Matemática Iberoamericana

Comparison of the classical BMO with the BMO spaces associated with operators and applications

Donggao Deng , Xuan Thinh Duong , Adam Sikora , and Lixin Yan

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Abstract

Let $L$ be a generator of a semigroup satisfying the Gaussian upper bounds. A new ${\rm BMO}_L$ space associated with $L$ was recently introduced in [Duong, X. T. and Yan, L.: {New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications}. \textit{Comm. Pure Appl. Math.} {\bf 58} (2005), 1375-1420] and [Duong, X. T. and Yan, L.: {Duality of Hardy and BMO spaces associated with operators with heat kernels bounds}. \textit{J. Amer. Math. Soc.} {\bf 18} (2005), 943-973]. We discuss applications of the new ${\rm BMO}_L$ spaces in the theory of singular integration. For example we obtain ${\rm BMO}_L$ estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space ${\rm BMO}_L$ might coincide with or might be essentially different from the classical BMO space.

Article information

Source
Rev. Mat. Iberoamericana, Volume 24, Number 1 (2008), 267-296.

Dates
First available in Project Euclid: 16 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1216247102

Mathematical Reviews number (MathSciNet)
MR2435973

Zentralblatt MATH identifier
1283.42036

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory 47B38: Operators on function spaces (general)

Keywords
BMO space Hardy space Dirichlet and Neumann Laplacians semigroup Gaussian bounds fractional powers purely imaginary powers spectral multiplier

Citation

Deng , Donggao; Duong , Xuan Thinh; Sikora , Adam; Yan , Lixin. Comparison of the classical BMO with the BMO spaces associated with operators and applications. Rev. Mat. Iberoamericana 24 (2008), no. 1, 267--296. https://projecteuclid.org/euclid.rmi/1216247102


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References

  • Alexopoulos, G.: Spectral multipliers for Markov chains. J. Math. Soc. Japan 56 (2004), no. 3, 833-852.
  • Auscher, P., Duong, X. T. and McIntosh, A.: Boundedness of Banach space valued singular integral operators and Hardy spaces. Preprint, 2004.
  • Auscher, P. and Russ, E.: Hardy spaces and divergence operators on strongly Lipschitz domain of $\mathbb R^n$. J. Funct. Anal. 201 (2003), 148-184.
  • Auscher, P., Russ, E. and Tchamitchian, P.: Hardy Sobolev spaces on strongly Lipschitz domain of $\mathbb R^n$. J. Funct. Anal. 218 (2005), 54-109.
  • Auscher, P. and Tchamitchian, P.: Square root problem for divergence operators and related topics. Astérisque 249. Soc. Math. France, 1998.
  • Chang, D.-C., Krantz, S. G. and Stein, E. M.: $H^p$ theory on a smooth domain in $\mathbb R^N$ and elliptic boundary value problems. J. Funct. Anal. 114 (1993), 286-347.
  • Christ, M.: $L\sp p$ bounds for spectral multipliers on nilpotent groups. Trans. Amer. Math. Soc. 328 (1991), 73-81.
  • Coifman, R. and Weiss. G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), 569-645.
  • Coulhon, T. and Duong, X. T.: Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv. Differential Equations 5 (2000), 343-368.
  • Cowling, M. and Meda, S.: Harmonic analysis and ultracontractivity. Trans. Amer. Math. Soc. 340 (1993), 733-752.
  • Davies, E. B.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92. Cambridge Univ. Press, Cambridge, 1989.
  • Duong, X. T. and McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoamericana 15 (1999), 233-265.
  • Duong, X. T., Ouhabaz, E. M. and Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196 (2002), 443-485.
  • Duong, X. T., Ouhabaz, E. M. and Yan, L.: Endpoint estimates for Riesz transforms of magnetic Schrödinger operators. Ark. Mat. 44 (2006), 261-275.
  • Duong, X. T. and Yan, L.: New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications. Comm. Pure Appl. Math. 58 (2005), 1375-1420.
  • Duong, X. T. and Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernels bounds. J. Amer. Math. Soc. 18 (2005), 943-973.
  • Duong, X. T. and Yan, L.: On commutators of fractional integrals. Proc. Amer. Math. Soc. 132 (2004), 3549-3557.
  • Dziubański, J., Garrigós, G., Martínez, T., Torrea, J. L. and Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249 (2005), 329-356.
  • Fefferman, C. and Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193.
  • Gunawan, H.: On weighted estimates for Stein's maximal function. Bull. Austral. Math. Soc. 54 (1996), 35-39.
  • Hebisch, W.: A multiplier theorem for Schrödinger operators. Colloq. Math. 60/61 (1990), no. 2, 659-664.
  • John, F. and Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961) 415-426.
  • Martell, J. M.: Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Studia Math. 161 (2004), 113-145.
  • Mauceri, G. and Meda, S.: Vector-valued multipliers on stratified groups. Rev. Mat. Iberoamericana 6 (1990), 141-154.
  • McIntosh, A.: Operators which have an $H_\infty$ functional calculus. In 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.
  • Müller, D. and Stein, E. M.: On spectral multipliers for Heisenberg and related groups. J. Math. Pures Appl. (9) 73 (1994), 413-440.
  • Sikora, A. and Wright, J.: Imaginary powers of Laplace operators. Proc. Amer. Math. Soc. 129 (2001), 1745-1754
  • Stein, E. M.: Singular integral and differentiability properties of functions. Princeton Mathematical Series 30. Princeton University Press, Princeton, N.J., 1970.
  • Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals. Princeton Mathematical Series 43. Princeton Univ. Press, 1993.
  • Strauss, W. A.: Partial differential equation: An introduction. John Wiley& Sons, Inc., New York, 1992.
  • Torchinsky, A.: Real-variable methods in harmonic analysis. Pure and Applied Mathematics 123. Academic Press, Inc. Orlando, FL, 1986.
  • Triebel, H.: Theory of function spaces. Monographs in Mathematics 78. Birkhauser verlag, Basel, Boston, Stuggart, 1983.
  • Uchiyama, A. and Wilson, J. M.: Approximate identities and $H^1(\mathbb R)$. Proc. Amer. Math. Soc. 88 (1983), 53-58.
  • Varopoulos, N., Saloff-Coste, L. and Coulhon, T.: Analysis and geometry on groups. Cambridge Tracts in Mathematics 100. Cambridge Univ. Press, London, 1992.
  • Weiss, G.: Some problems in the theory of Hardy spaces. In Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, 1978), Part 1, 189-200. Proc. Sympos. Pure Math. 35. Amer. Math. Soc., Providence, R.I., 1979.