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1995 | 113 | 2 | 177-196
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A sharp correction theorem

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Under certain conditions on a function space X, it is proved that for every $L^∞$-function f with $∥f∥_{∞} ≤ 1$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, $mes{φ ≠ 1} ≤ ɛ∥f∥_1$ and $∥φf∥_X ≤ const(1 + log ɛ^{-1})$. For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of $L^∞$-functions on $ℝ^n$ whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.
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Twórcy
  • Steklov Mathematical Institute, St. Petersburg Branch (POMI), Fontanka 27, 191011 St. Petersburg, Russia
Bibliografia
  • [1] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985.
  • [2] S. V. Khrushchëv, Simple proof of a theorem on removable singularities of analytic functions satisfying a Lipschitz condition, Zap. Nauchn. Sem. LOMI 113 (1981), 199-203 (in Russian); English transl.: J. Soviet Math. 22 (1983), 1829-1832.
  • [3] S. V. Khrushchëv [S. V. Hruščëv] and S. A. Vinogradov, Free interpolation in the space of uniformly convergent Taylor series, in: Lecture Notes in Math. 864, Springer, Berlin, 1981, 171-213.
  • [4] S. V. Kisliakov, Once again on the free interpolation by functions which are regular outside a prescribed set, Zap. Nauchn. Sem. LOMI 107 (1982), 71-88 (in Russian); English transl. in J. Soviet Math.
  • [5] S. V. Kisliakov, Quantitative aspect of correction theorems, Zap. Nauchn. Sem. LOMI 92 (1979), 182-191 (in Russian); English transl. in J. Soviet Math.
  • [6] S. V. Kisliakov, Quantitative aspect of correction theorems, II, Zap. Nauchn. Sem. POMI 217 (1994), 83-91 (in Russian).
  • [7] J. Rubio de Francia, F. J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), 7-48.
  • [8] S. A. Vinogradov, A strengthened form of Kolmogorov's t heorem on the conjugate function and interpolation properties of uniformly convergent power series, Trudy Mat. Inst. Steklov. 155 (1981), 7-40 (in Russian); English transl.: Proc. Steklov Inst. Math. 155 (1981), 3-37.
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