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Under certain conditions on a function space X, it is proved that for every $L^{\infty }$-function f with $\parallel f{\parallel }_{\infty }\le 1$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, $mes\phi \ne 1\le \varepsilon \parallel f{\parallel }_1$ and $\parallel \phi f{\parallel }_X\le const(1+log{\varepsilon }^{-1})$. 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.