I was wondering, if it is somehow possible to add a TextBox in the Application Bar area (the same way the Internet Explorer does), but I didn't find any information source about this. Have you any idea, how to accomplish this task?
You will not be able to add UIElements inside ApplicationBar as it is not derived from UIElement.
Alternatively, you can try adding a Grid at the bottom of the screen and color it similar to the ApplicationBar. Then add a TextBlock inside the Grid. This is just a workaround and not recommended as it is not according to the standards.
You can also check the following link: http://www.maxpaulousky.com/blog/archive/2011/01/10/bindable-application-bar-extensions-for-windows-phone-7.aspx