The question of how the Arabs conquered their massive empire is one that has been repeatedly asked and answered in this subreddit. It is also known that most of the regions under their control did not become majority-Muslim until around the 11th century, or nearly 400 years after the initial conquests. Before that time, these territories would have been majorly Christian or Zoroastrian, and so have regarded their Arab rulers as hostile occupiers. My question is: how did the Arabs maintain control over a hostile population for so long that they converted? In particular, the Christian populations of the erstwhile Byzantine lands would have been very receptive to European Christians trying to stir up rebellions, isn't it?
This is an interesting question, but I think it comes from an outdated understanding of early Islamic history.
Before that time, these territories would have been majorly Christian or Zoroastrian, and so have regarded their Arab rulers as hostile occupiers.
Why would the local peoples inherently have viewed their Arab rulers as hostile occupiers? Sometimes it is too easy to impose our own ideals of nationalism upon past peoples. Fred Donner in Muhammad and the Believers: At the Origins of Islam has even argued that early Islam was not a separate religion per se, but instead stressed a sort of monotheistic religious revival which appealed to many. In an edited book by Donner and Borrut titled Christians and Others in the Umayyad State, the lack of animosity and strict religious divisions is highlighted. There were many Christians in Umayyad service, and a system of emphasizing merit and personal connections took precedent over any strict religious identification division. Persian notables also openly supported the early Caliphate as much as early Christians. Non-Muslims were in the Caliphate military, often in important roles. Religious identity mattered, but far less than traditional explanations of the early Caliphate stress.
Even beyond the Umayyads, think about Tayeb El-Hibri's Reinterpreting Islamic Historiography, which challenges the traditional interpretation of the coming of the Abbasids as a Persian takeover of the previously-Arabic state. Instead, recent evidence suggests that such Persian-Abbasid alignment was developed much later and reapplied to the past by Islamic historians after Persian-Arab tensions developed.