Let \(\left( \Omega ,\Sigma ,\mu \right) \) be a \(\sigma \)-finite continuous measure space, \(S\left( \mu \right) \) the set of measurable functions on \( \Omega \) and \(M^{\tau }\) the Marcinkiewicz space corresponding to the function \(\varphi \left( s\right) =s^{\tau }\), \(0\leq \tau <1\). The main result from the paper is:
{Theorem 1}. Assume that a pair of numbers \(0\leq \tau _{0}<\tau _{1}\leq 1\), a Banach space \(Y\), and a quasilinear operator \(T:Y\rightarrow S\left( \mu \right) \) are fixed. Assume also that a function \(\xi \left( \tau \right) \) is given, and this function does not increase in \(\tau \) and is equivalent to a logarithmically convex function. For \(t\in \left( 0,1\right) \), define the function \(\psi _{\xi }\left( t\right) =\sup\limits_{\tau \in \left( \tau _{0},\tau _{1}\right) }\frac{t^{\tau }}{ \xi \left( \tau \right) }\). For the inequality \(\left\| T\mid Y\rightarrow M^{\tau }\right\| \leq c\xi \left( \tau \right) \) to be satisfied for all \(\tau \in \left( \tau _{0},\tau _{1}\right) \), it is necessary and sufficient that the inequality \[ \sup\limits_{t\in \left( 0,1\right) }\frac{\psi _{\xi }\left( t\right) }{t} \int_{0}^{t}\left( Tx\right) ^{\ast }\left( s\right) ds\leq c\left\| x\mid Y\right\| \] is satisfied for all \(x\in Y\).
Two applications of this result are given.