Maximum a posteriori estimators in $\ell^p$ are well-defined for diagonal Gaussian priors

We prove that maximum a posteriori estimators are well-defined for diagonal Gaussian priors $\mu$ on $\ell^p$ under common assumptions on the potential $\Phi$. Further, we show connections to the Onsager--Machlup functional and provide a corrected and strongly simplified proof in the Hilbert space case $p=2$, previously established by Dashti et al (2013) and Kretschmann (2019). These corrections do not generalize to the setting $1 \leq p < \infty$, which requires a novel convexification result for the difference between the Cameron--Martin norm and the $p$-norm.