Nonclassical interpolation in spaces of smooth functions

Vladimir Ovchinnikov

Studia Mathematica (1999)

  • Volume: 135, Issue: 3, page 203-218
  • ISSN: 0039-3223

Abstract

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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 B M O 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.

How to cite

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Ovchinnikov, Vladimir. "Nonclassical interpolation in spaces of smooth functions." Studia Mathematica 135.3 (1999): 203-218. <http://eudml.org/doc/216651>.

@article{Ovchinnikov1999,
abstract = {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.},
author = {Ovchinnikov, Vladimir},
journal = {Studia Mathematica},
keywords = {bounded mean oscillation; inhomogeneous BMO-space; Bessel-potential; non-classical interpolation method},
language = {eng},
number = {3},
pages = {203-218},
title = {Nonclassical interpolation in spaces of smooth functions},
url = {http://eudml.org/doc/216651},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Ovchinnikov, Vladimir
TI - Nonclassical interpolation in spaces of smooth functions
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 3
SP - 203
EP - 218
AB - 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.
LA - eng
KW - bounded mean oscillation; inhomogeneous BMO-space; Bessel-potential; non-classical interpolation method
UR - http://eudml.org/doc/216651
ER -

References

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  1. [1] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. Zbl0344.46071
  2. [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. [3] Z. Ciesielski and T. Figiel, Spline bases in classical function spaces on compact manifolds, Studia Math. 76 (1983), 1-58. Zbl0599.46041
  4. [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. Zbl0395.46030
  5. [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. [6] V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Reports 1 (1984), 349-516. Zbl0875.46007
  7. [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. [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. Zbl0915.46066
  9. [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. [10] A. A. Sedaev, Description of interpolation spaces of the pair ( L α p 0 , L α p 1 ) and some related questions, Dokl. Akad. Nauk SSSR 209 (1973), 798-800 (in Russian); English transl. in Soviet Math. Dokl. 14 (1973). 
  11. [11] C. Segovia and R. L. Wheeden, Fractional differentiation of the commutator of the Hilbert transform, J. Funct. Anal. 8 (1971), 341-359. Zbl0234.44007
  12. [12] R. S. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539-558. Zbl0437.46028
  13. [13] R. S. Strichartz, Traces of BMO-Sobolev spaces, Proc. Amer. Math. Soc. 83 (1981), 509-513. Zbl0474.46024
  14. [14] H. Triebel, The Theory of Function Spaces, Birkhäuser, Basel, 1983. Zbl0546.46028

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