I am looking for a nice, clean and proper derivation of the following statement:
Given:
$\arg\min_{f \in\mathcal{H}_K}\{||f-f_{p}||_p^2 +\lambda||f||^2_K\}$
where
$f_{p}(x) = \int_Y ydp(y|x)$,
$L_K(f)(x) := \int_X K(x,t)f(t)dp_X(t)$
$||f||_p^2 = \int_X |f(x)|^2dp_X(x)$
$H_K$ is some Hilbert space (RKHS) with kernel $K$. $p_X$ is marginal probability measure of $X$ and $p(y|x)$ is a conditional probability measure.
The solution is given by
$f_{\lambda} = (L_K + \lambda I)^{-1}L_K f_p$
I can only come up with handwavy ways to incorporate $L_K$, so I would appreciate help.I am quite new to functional analysis but not to regression.