设函数 $f(u, v)$ 具有二阶连续的偏导数，$z=f\left(\frac{x^2+y^2}{2}, \frac{x^2-y^2}{2}\right)$ 满足 $y^2 \frac{\partial^2 z}{\partial x^2}-x^2 \frac{\partial^2 z}{\partial y^2}-\frac{y^2}{x} \frac{\partial z}{\partial x}+$ $\frac{x^2}{y} \frac{\partial z}{\partial y}=4 y^2$,
（I）求 $\frac{\partial^2 f}{\partial u \partial v}$ ；
（II）若 $\left.\frac{\partial f}{\partial u}\right|_{(u, u)}=\ln |2 u|-u, f(v, v)=2 v \ln |2 v|-v-\frac{v^2}{2}$ ，求 $f(1,1)$ ．