How mature is AJAX support for mobile phones? Are there mobile versions of some famous AJAX frameworks? I would live even with a limited support for some ajax lib.
iPhones won't cache anything larger than 25 KB uncompressed, which means libraries like jQuery and Prototype won't be cached like they would be on a normal computer. I suspect this sort of thing is the case with lots of other mobile browsers, too.
XUI is a slimmed down jQuery-like library that's fairly popular.
Mobile phone market is very fragmented and you need to restrict yourself to a specific technology to get a useful answer. As far as I know there is no cross platform mobile framework available. However some mobile browsers are quite advanced and they will be able to handle jQuery or other libraries. I have seen jQuery run on S60 built in browser as well as Opera (not mini) at the same platform. I have also successfully run jQuery in mobile IE (Windows Mobile Professional Edition). However I wouldn't say it is safe to use jQuery on these platforms. You also want to ask the question regarding the size of the downloaded library as well as the execution speed.
I think mobile ajax is not mature yet, but you should read this wmlprogramming thread, and check this preliminar version of dojo mobile too.
Peter-Paul Koch (ppk) started working on mobile device compatibility tests and has made compatibility charts for mobile devices.
Once these are documented, libraries authors can begin to abstract the differences. I don't know of many libraries that are cross-platform yet, but Peter's work should speed development.