From what I understand there are no known primary historical sources on Muhammad. Does the same apply to his companions such as Abu Bakr and Umar, the first two Caliphs?

by ManInBlack94

For example are there any historical records on Umar ibn Al-Khattab taking Jerusalem away from Byzantine?

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There is an inscription that records the death of 'Umar and is dated to 24 AH (644 CE), which conforms to the date known from Muslim sources. This inscription is sometimes referred to as the "Zuhayr inscription", named after its author. Regarding sources for Muhammad, his name is mentioned in near-contemporary sources; for example, there is the account of Sebeos.