There are multiple answers out there as to why hiding Luke from his father on his father's home planet makes sense (the memories are too painful for Vader, Vader had no idea he had a son, and so on...). But what I want to know is, why wasn't anyone worried about the Emperor? He has no negative associations with Tatooine, knows about Vader's children, is incredibly clever, and is powerful enough in the Force to defeat Yoda.
There's a couple of reasons why
Palpatine bought that Padme was killed by Vader
We see this in the direct dialogue of this scene (about 4:40)
VADER: Where is Padme? Is she safe?
PALPATINE: It seems, in your anger, you killed her
And remember, her funeral made it seem as if she died pregnant
It's entirely possible he knew and wanted to keep Vader in the dark here, but it seems unlikely because...
Pursuing them would have been difficult
Vader had his hands deep in the Empire. While the Emperor didn't share everything with him, a long-running search on the Outer Rim without any stated purpose would have seemed odd sooner or later. And even then, Obi-Wan knew how to go to ground and disappear, as did Yoda (each would have been a far more tempting target to kill).
Therefore, Luke was not really hidden from Empire and Emperor, he merely lived on backwater planet with his relatives. Even Kenobi was not really hiding, he just changed his first name (Obi-Wan to Ben). All of this implies that originally Emperor, Empire or Vader didn't even supposed to know about Tatooine, and only chance (or will of the Force) brought them there.