We prove that maximum a posteriori estimators are well-defined for diagonal Gaussian priors ,u on tp under common assumptions on the potential F. 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 Inverse Problems 29 095017); Kretschmann (2019 PhD Thesis). These corrections do not generalize to the setting 1 ? p < 8, which requires a novel convexification result for the difference between the Cameron-Martin norm and the p-norm.
