A sharp correction theorem

Under certain conditions on a function space X, it is proved that for every ${\displaystyle L^{\infty }}$-function f with ${\displaystyle \parallel f{\parallel }_{\infty }\le 1}$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, ${\displaystyle mes\{\phi \ne 1\}\le \varepsilon \parallel f{\parallel }_1}$ and ${\displaystyle \parallel \phi f{\parallel }_X\le const(1+{\log \varepsilon }^{-1})}$. For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of ${\displaystyle L^{\infty }}$-functions on ${\displaystyle {ℝ}^n}$ whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.