AjaxSwing does
not let you run Swing applications in the browser. It
converts the app into HTML and JavaScript that runs in the browser. Not the same thing at all.
If that's what you want, that's what GWT does as well, but for free.
If you want to run
Java in the browser, that's what
Applets are for.
If you want pull Swing apps to client machines for execution, that's what Java Web Start is all about.