In the earlier days of the Empire, as Christianity rose, were Christians in (say SW Anatolia) persecuted as badly as Jews in Palestine?
I know that Diocletian was particularly brutal toward the Christians, so when did it die down? Was it before Constantine converted (I know his mother became Christian prior to him). Or did it happen after?
It's really difficult to compare persecutions, especially in a case like this where the persecutions were really quite different. Persecution of Jews occurred because of gradual Roman intrusions on Judean political autonomy, causing a revolt in 70. A lot of Jews were killed in a revolt in the early 2nd century, too, but the motives are unclear, and there's no reason to assume it was due to Roman persecution. It may've been leftover tensions in Egypt between Jews and Greeks, which had existed long-term and was part of precipitating the revolt in 70, too. In fact, the Roman government often had the role of protecting Jewish religious worship against locals in the Mediterranean who did not respect it.
The revolt in 135 was also incredibly damaging, and seems to have been due to actions of the Roman government. This may've been a ban on circumcision, or it may've been an attempt to re-found Jerusalem as a Greco-Roman city, complete with a temple to Jupiter. Both of those things happened, and both are mentioned by our sources in the context of setting off the revolt--it's just unclear which one set it off and which one was a punishment. But the big damage to Jews from the Romans was mostly political in cause, at least in 2/3 serious revolt-quashings.
Christian persecution may've originated from its refusal to associate with the imperial cult but lacking from the long-term recognition and acceptance Judaism had doing just that. But exactly what the persecution of Christianity consisted of is unclear, and was not consistently enforced in the empire. And the fact that Christians weren't in the political position to be rebelling against Rome meant that things looked different for the two, enough to make comparison challenging.
source:
Smallwood, E. Mary. The Jews under Roman Rule: From Pompey to Diocletian. Leiden: Brill, 1976.