Acts 1:7 YLT and he said unto them, 'It is not yours to know times or seasons that the Father did appoint in His own authority.
What does "in His own authority" mean in this verse?
The simple answer is that the phrase addresses the nature of the authority behind the establishment of the times and seasons just mentioned. It tells us that these have come from God Himself.
A more complete answer follows from a careful examination of the Greek.
The original Greek for that expression says:
ἐν τῇ ἰδίᾳ ἐξουσίᾳ
The first word "ἐν" is a preposition meaning by, in, on, or among. It is rather straightforward and unsurprising here. The next word, "τῇ" is the feminine form of the definite article, which means it refers to a feminine noun. The noun, of course, is "ἐξουσίᾳ", which is feminine, meaning "authority." The word "ἰδίᾳ" just before it is a feminine singular adjective, in the Greek dative case, for the noun. Because it is dative, it would usually be understood as a marker for the indirect object. With the "authority" being that indirect object, the direct object would be "the times or seasons," and "the Father" is the one doing the action.
In English, the translation would usually place this phrase in a prepositional form rather than as an indirect object. We would say "by His own authority." It comes across as emphasizing by what authority something was done: in this case, the appointment of the times and seasons.