None at all. Palm split into two companies in 2003, PalmOne and PalmSource. PalmOne handled the hardware and PalmSource handled the software. BeOS went with PalmSource.
PalmOne eventually bought the full rights to the Palm trademark from PalmSource in 2005 and switched back to being Palm. This is the company that made WebOS and got bought by HP.
PalmSource got bought by ACCESS (the company that made NetFront, one of the early mobile webbrowsers) and the rights to BeOS went to them in the sale.
I think that Access provides the web browser in the Kindle. That's probably the most widely-seen product by the most people.
It also tried to made a multitasking multimedia-capable PalmOS, PalmOS 6 "Cobalt."
https://www.palmsource.com/palmos/cobalt.html
My submission of it here got some discussion 9mth ago: