Vader and Palpatine knew Obi-wan was with Padme on Mustafar. They knew Padme (dead or unconscious) was laying near the landing platform, before Vader and Obi-wan duelled. They knew that her body was gone from Mustafar when Palpatine appeared. And finally they knew that her body appeared on Naboo for her funeral.
My question is, why didn't they track down who delivered the body to Naboo? Obviously, this person would have some relations with Obi-wan, because he was the most likely suspect to remove Padme's body from Mustafar before Palpatine arrived. I suppose Palpatine did have connections on his home planet to help him unravel this mystery.