Multipliers for the Calderón–Lozanovskii construction

Using a new approach for the Calderón–Lozanovskii construction $\varphi (X, L^{\infty})$ involving an arbitrary ideal space $X$, a Lebesgue space $L^{\infty}$, and a concave function $\varphi$, an exact description of the multiplier space $M(\varphi_0 (X, L^{\infty}) \to \varphi_1 (X, L^{\infty}))$ is given, provided that the ratio ${{\varphi_0(\cdot, 1)} /{\varphi_1(\cdot, 1)}}$ does not increase. Namely, it is shown that the equality
$$ M(\varphi_0 (X, L^{\infty}) \to \varphi_1 (X, L^{\infty}))=\varphi_2 (X, L^{\infty}) $$
is satisfied, where the function $\varphi_2 $ is determined constructively from the functions $\varphi_0, \varphi_1$. The absence of restrictions on the ideal space $X$ and the exact description of the function $\varphi_2 $ enables us to apply the results thus obtained to a wide class of ideal spaces that are not symmetric and cannot be reduced to symmetric ones by an introduction of weight functions, for example, Morrey spaces.