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A sharp correction theorem

S. Kisliakov — 1995

Studia Mathematica

Under certain conditions on a function space X, it is proved that for every $L^{\infty}$ -function f with $∥ f ∥_{\infty} \leq 1$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, $m e s φ \neq 1 \leq ɛ ∥ f ∥_{1}$ and $∥ φ f ∥_{X} \leq c o n s t (1 + l o g ɛ^{- 1})$ . For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of $L^{\infty}$ -functions on $ℝ^{n}$ whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.

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