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1999 | 135 | 3 | 203-218
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Nonclassical interpolation in spaces of smooth functions

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We prove that the fractional BMO space on a one-dimensional manifold is an interpolation space between C and $C^1$. We also prove that $BMO^1$ is an interpolation space between C and $C^2$. The proof is based on some nonclassical interpolation constructions. The results obtained cannot be transferred to spaces of functions defined on manifolds of higher dimension. The interpolation description of fractional BMO spaces is used at the end of the paper for the proof of the boundedness of commutators of the Hilbert transform.
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Twórcy
  • Mathematical Department, Voronezh State University, Universitetskaia pl. 1, 394693, Voronezh, Russia, vio@func.vsu.ru
Bibliografia
  • [1] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.
  • [2] A. P. Calderón, Commutators, singular integrals on Lipschitz curves and applications, in: Proc. Internat. Congress Math. Helsinki, 1978, Vol. 1, 1980, 85-96.
  • [3] Z. Ciesielski and T. Figiel, Spline bases in classical function spaces on compact manifolds, Studia Math. 76 (1983), 1-58.
  • [4] B. Mitiagin and E. M. Semenov, Absence of interpolation of linear operators in spaces of smooth functions, Izv. Akad. Nauk SSSR 41 (1977), 1229-1266 (in Russian); English transl. Math. USSR-Izv. 11 (1977), 1289-1328.
  • [5] V. I. Ovchinnikov, Interpolation theorems, resulting from the Grothendieck inequality, Funktsional. Anal. i Prilozhen. 10 (1976), no. 4, 45-54 (in Russian); English transl.: Functional Anal. Appl. 10 (1977), 287-294.
  • [6] V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Reports 1 (1984), 349-516.
  • [7] V. I. Ovchinnikov, Interpolation properties of fractional BMO space, in: All-Union School on the Theory of Operators in Functional Spaces, Kuῐbyshev, 1988, 142 (in Russian).
  • [8] V. I. Ovchinnikov, On reiteration theorems, in: Voronezh Winter Mathematical Schools, P. Kuchment and V. Lin (eds.), Amer. Math. Soc., Providence, R.I., 1998, 185-198.
  • [9] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., Providence, R.I., 1986.
  • [10] A. A. Sedaev, Description of interpolation spaces of the pair $(L^p_α_0, L^p_α_1)$ and some related questions, Dokl. Akad. Nauk SSSR 209 (1973), 798-800 (in Russian); English transl. in Soviet Math. Dokl. 14 (1973).
  • [11] C. Segovia and R. L. Wheeden, Fractional differentiation of the commutator of the Hilbert transform, J. Funct. Anal. 8 (1971), 341-359.
  • [12] R. S. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539-558.
  • [13] R. S. Strichartz, Traces of BMO-Sobolev spaces, Proc. Amer. Math. Soc. 83 (1981), 509-513.
  • [14] H. Triebel, The Theory of Function Spaces, Birkhäuser, Basel, 1983.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv135i3p203bwm
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