I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are Orlicz spaces and spaces $M(A), M(B)$ are Lorentz spaces corresponding to $L_A, L_B$ respectively.
Set $\phi_K(t)=1/K^{-1}(1/t)$, for $K=A,B,C$ are concave non-decreasing functions.
And I would like to understand whether the Young's condition $$ s\phi_C(s)\leq c\phi_A(s)\phi_B(s) $$ is sufficient to guarantee that $$ M(A)\ast M(B) \subset \Lambda(C), $$ where $\Lambda(C)$ is a Banach space of all locally integrable functions such that $\int_0^{\infty} f^*(s)d\phi_A(s)$ is finite.
