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Consider the equation
where ƒ ∈ Lp(ℝ), p ∈ (1, ∞) and
By a solution of (*), we mean any function y absolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the space Lp(ℝ), i.e.
(1) for any function 
, there exists a unique solution y ∈ Lp(ℝ) of (*);
(2) there exists an absolute constant c1(p) > 0 such that the solution y ∈ Lp(ℝ) of (*) satisfies the inequality
We study the following problem on the strengthening estimate (**). Let a non-negative function 
be given. We have to find minimal additional restrictions for θ under which the following inequality holds:
Here, y is a solution of (*) from the class Lp(ℝ), and c2 (p) is an absolute positive constant.
