I meant, which one should I choose in this sentence?"On my authority" suggests "with my approval"; "under my authority" (if we used it at all) might be taken to mean something vaguer, like "while operating as my subordinate."
We can help you more if you tell us what you want the sentence to mean.
Check this:Re-read The Newt's post. Did 'he' do it with 'my' approval, or did 'he' do it while operating as 'my' subordinate? Without knowing what you want to say, it's impossible to say which one you should choose.
Who is 'he', and what relationship does he have with 'me'. What did he do? And why?