If you are getting inundated with fan comments whenever you send a message from your community page or the Artist App, then congratulations for having such vocal and passionate fans.
We do our best to consolidate new comments into digest emails, but if you are still finding that your inbox is overflowing, then you can disable email notifications via Settings > Profile and unticking the “Email me when fans comment on my messages” checkbox. You can also control push notifications on the Account tab in the Artist App.