On the $\mathcal{L}$-Characteristic of the Superposition Operator in Lebesgue Spaces with Mixed Norm

Abstract

We consider the superposition operator $Fx(t,s)=f(t,s,x(t,s))$ of functions of two variables in spaces with mixed norm $[L_p\to L_q]$. After establishing a necessary and sufficient acting condition, we get some conclusions on the $\mathcal{L}$-characteristic of $F$. We also prove some theorems, which imply that $F$ is uniformly continuous on balls in the interior of its $\mathcal{L}$-characteristic.