Jesus was born, um, somewhere around this time

[Justtristo]

Now many people came in crowds to him, for they were greatly moved by his words. Herod, who feared that the great influence John had over the masses

It would mean that Antipas executed someone who was a subject of the Roman Prefect of Judaea. The Romans might well have said "go ahead" but the passage in question does not even indicate that Antipas asked. He knew his limitations as a client-king. When he wanted to marry Herodias he sailed all the way to Rome to get Tiberius' permission. More to the point, Josephus knew Antipas' limitations as a client-king.

The gospel writers, OTOH, don't seem to have known shit.

Herod Antipas did authority over the Galilee and Perea even despite being limited as a client king, that would assume John the Baptist came from either one of those areas.