Let $g:\mathbb Q_p^n\longrightarrow{\mathbb Z}$ increasing and radial. We define the following operator
$$Hf(x)=\frac{1}{p^{n\left|{g(x)}\right|}}\displaystyle\int_{B_{g(x)}^n}\left |{f(y)}\right |dy$$
where $p$ is a fixed prime number and $B_{g(x)}^n$ is a ball with center at zero and radius $p^{g(x)}$. I'm trying to prove if it operator is self-adjoint in $L^2(\mathbb Q_p^n)$. it's a Hardy-like operator. $\mathbb Q_p$ is the field of p-adic numbers.
\begin{align*} \left<{Hf,h}\right>&=\int_{\mathbb Q}\frac{1}{p^{\left|{g(x)}\right|}}\displaystyle\int_{B_{g(x)}}\left |{f(y)}\right |dy \ \overline{h(x)}dx\\ &=\int_{\mathbb Q}\frac{1}{p^{\left|{g(x)}\right|}}\displaystyle\int_{B_{g(x)}}\left |{f(y)}\right | \overline{h(x)}dydx\\ &=\int_{\mathbb Q}\frac{1}{p^{\left|{g(x)}\right|}}\displaystyle\int_{B_{g(x)}}\left |{f(x)}\right | \overline{h(y)}dydx\\ &=\int_{\mathbb Q}\left |{f(x)}\right |\overline{\frac{1}{p^{\left|{g(x)}\right|}}\displaystyle\int_{B_{g(x)}}h(y)dy}dx\\ &=\left<{f,Hh}\right> \end{align*}
I have the doubt with the exchange of variables that i did and the module of $f$.