After received my PhD, can I publish a sole-author paper for some extended results?
You don't need to ask permission for your ongoing work. It is entirely appropriate to extend what you have done and publish it. You should be sure to cite the older work as needed to avoid questions of self-plagiarism, of course. 

An acknowledgement of your advisor for their past help is a good and appropriate thing to do, but not necessarily required. I would do it unless there were some specific reason not to do so. 

Asking them if they want to participate can also be good, but not at all necessary. It is good simply because having collaborative relationships is good. If you decide to do that, then co-authorship would be a consideration, of course.  But sole-authorship is also proper and is eventually the place (nearly) every researcher will wind up.