We consider the equationwhereUnder these conditions, (1) is correctly solvable in L1(ℝ), i.e.(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimatewhere c2 is some absolutely positive constant.