Adam Clarke on Ezra 1:2

The Lord God of heaven - It is not unworthy of remark, that in all the books written prior to the captivity, Jehovah is called The Lord of Hosts; but in all the books written after the captivity, as 2 Chronicles, Ezra Nehemiah, and Daniel, he is styled The God of Heaven. The words however have the same meaning.

All the kingdoms of the earth. At this time the empire of the Medo-Persians was very extensive: according to ancient writers, Cyrus, at this time, reigned over the Medes, Persians, Hyrcanians, Armenians, Syrians, Assyrians, Arabians, Cappadocians, Phrygians, Lydians Phoenicians, Babylonians, Bactrians, Indians, Saci, Cilicians, Paphlagonians, Moriandrians, and many others. His empire extended on the East, to the Red Sea; on the North, to the Euxine Sea; on the West, to the island of Cyprus and Egypt; and on the South, to Ethiopia.