A sharp correction theorem

S. Kisliakov

Studia Mathematica (1995)

  • Volume: 113, Issue: 2, page 177-196
  • ISSN: 0039-3223

Abstract

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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, m e s φ ≠ 1 ≤ ɛ ∥ f ∥ 1 and ∥ φ f ∥ X ≤ c o n s t ( 1 + l o g 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.

How to cite

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Kisliakov, S.. "A sharp correction theorem." Studia Mathematica 113.2 (1995): 177-196. <http://eudml.org/doc/216168>.

@article{Kisliakov1995,
abstract = {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.},
author = {Kisliakov, S.},
journal = {Studia Mathematica},
keywords = {Lipschitz functions; Fourier sums; convolutions; Calderón-Zygmund kernels},
language = {eng},
number = {2},
pages = {177-196},
title = {A sharp correction theorem},
url = {http://eudml.org/doc/216168},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Kisliakov, S.
TI - A sharp correction theorem
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 2
SP - 177
EP - 196
AB - 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.
LA - eng
KW - Lipschitz functions; Fourier sums; convolutions; Calderón-Zygmund kernels
UR - http://eudml.org/doc/216168
ER -

References

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  1. [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. [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. Zbl0476.30034
  3. [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. [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. Zbl0499.41002
  5. [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. Zbl0434.42017
  6. [6] S. V. Kisliakov, Quantitative aspect of correction theorems, II, Zap. Nauchn. Sem. POMI 217 (1994), 83-91 (in Russian). 
  7. [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. Zbl0627.42008
  8. [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. Zbl0468.30036

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