In this paper, we define the superposition operator $P_g$ where $g:\mathbb{N}^2\times\mathbb R\to\mathbb R$ by $P_g((x_{ks}))=g(k,s,x_{ks})$ for all real double sequence $(x_{ks})$. Chew \& Lee [4] and Petranuarat \& Kemprasit [7] have characterized $P_g:l_p\to l_1$ and $P_g:l_p\to l_q$ where $1\leq p,q<\infty$, respectively. The main goal of this paper is to construct the necessary and sufficient conditions for the continuity of $P_g:\mathcal L_p\to\mathcal L_1$ and $P_g:L_p\to\mathcal L_q$ where $1\leq p,q<\infty$.