If user has a lot of emails which they need to delete here is a command which can help at the server end:
Search-Mailbox -Identity “User Name” -SearchQuery “Received:<$(’12/31/2013′)” -DeleteContent
ref: https://social.technet.microsoft.com/Forums/exchange/en-US/6114adb8-7e65-46e0-b54f-704ad15ef723/command-to-delete-all-the-items-from-a-mailbox-till-a-specific-date?forum=exchange2010