The next set of Palms running the next-gen OS will be based on linux. Access Linux Platform, as it is called, will be able to run older Palm apps via emulation and will also be designed around a 1-handed user interface. For the most part, however, most Palm applications will need to be re-written, some completely. Not all Palms will be able to run the new OS either:

Alas, ALP won't support single-chip phones - devices where the application processor runs the signalling stack - but Diaz said PalmSource recognised the industry was moving in this direction. "It's something we can do," he claimed.
ALP is running off of Linux 2.6.12 Kernel and will use highly optimized applications, and it will be interesting to see how a change in platform affects the popular handheld.