Let $T$ be the vector-valued integral operator defined by $$ \begin{eqnarray*} T &:&L^{2}(0,1)\times L^{2}(0,1)\longrightarrow L^{2}(0,1)\times L^{2}(0,1), \\ (u,v)^{t} &\mapsto &T(u,v)^{t}=\int_{0}^{1}K(s,x)(u,v)^{t}ds \\ &=&\int_{0}^{1}\left( \begin{array}{cc} K_{11}(s,x) & K_{12}(s,x) \\ K_{21}(s,x) & K_{22}(s,x)% \end{array}% \right) \left( \begin{array}{c} u(s) \\ v(s)% \end{array}% \right) ds \end{eqnarray*}$$ where $K_{i,j}$ are smooth enough. I want to find the norm or at least an optimal one of $T$ in the space $L^{2}(0,1)\times L^{2}(0,1)$ in function of $K_{i,j}$. My idea is to apply $K$ on both $(u,v)$ and bound the operator $T$ but it is very restrictive. \begin{eqnarray*} &&\left\Vert T(u,v)^{t}\right\Vert _{L^{2}(0,1)\times L^{2}(0,1)} \\ &=&\left\Vert \left( \int_{0}^{1}K_{11}(s,x)u(s)+K_{12}(s,x)v(s)ds,% \int_{0}^{1}K_{21}(s,x)u(s)+K_{22}(s,x)v(s)ds\right) \right\Vert _{L^{2}(0,1)\times L^{2}(0,1)} \\ &=&\max \left\{ \begin{array}{c} \left\Vert \int_{0}^{1}K_{11}(s,x)u(s)+K_{12}(s,x)v(s)ds\right\Vert _{L^{2}(0,1)} \\ ,\left\Vert \int_{0}^{1}K_{11}(s,x)u(s)+K_{12}(s,x)v(s)ds\right\Vert _{L^{2}(0,1)}% \end{array}% \right\} , \end{eqnarray*}
but \begin{eqnarray*} &&\left\Vert \int_{0}^{1}K_{11}(s,x)u(s)+K_{12}(s,x)v(s)ds\right\Vert _{L^{2}(0,1)} \\ &\leq &\left\Vert \int_{0}^{1}K_{11}(s,x)u(s)ds\right\Vert _{L^{2}(0,1)}+\left\Vert \int_{0}^{1}K_{12}(s,x)v(s)ds\right\Vert _{L^{2}(0,1)} \\ &\leq &\sqrt{\int_{0}^{1}\int_{0}^{1}K_{11}^{2}(s,x)dsdx}\left\Vert u\right\Vert _{L^{2}(0,1)}+\sqrt{\int_{0}^{1}\int_{0}^{1}K_{12}^{2}(s,x)dsdx}% \left\Vert v\right\Vert _{L^{2}(0,1)}..... \end{eqnarray*} and so on. I want to find an optimal way to estimate the norm of $T$ in an optimal way. Any ideas guys?. Thank you in advance.