Hardy operator and its adjoint

Question

For nonnegative function $f$ on $\mathbb R^n$, $$Hf(x)=\frac{1}{\Omega_n \left|x\right|^n} \int_{\left|y\right|<\left|x\right|}f(y)dy, \quad H^*f(x)= \int_{\left|y\right|\geq\left|x\right|}\frac{f(y)}{\Omega_n \left|y\right|^n}dy,\quad x\in\mathbb R^n\setminus\{0\},$$ where $\Omega_n$ is the volume of the unit ball in $\mathbb R^n$. How could I demonstrate that it is satisfied $$\int_{\mathbb R^n}g(x)Hf(x)dx=\int_{\mathbb R^n}f(x)H^*g(x)dx.$$ I have difficulties when trying to perform the change of integration order in order to demonstrate the equality.