What it means
の下で expresses that an action or situation takes place under the guidance, supervision, authority, or influence of a person, group, system, or condition. It commonly follows a noun and is often used in formal or written contexts. Depending on the noun, it can describe receiving guidance from someone, working within an institution's control, or being affected by a particular influence.
Usage note
の下で is usually read のもとで in this meaning and should not be confused with the literal location expression 下で, read したで, meaning physically under something. The related form の下に can describe a state or existence under someone's authority, whereas の下で emphasizes an action carried out under that authority or influence.
How to use it
経験豊かな監督の下で、選手たちは新しい戦術を学びました。
Under the guidance of an experienced coach, the players learned a new strategy.
経験豊かな監督の下で、選手たちは新しい戦術を学んだ。
Under the guidance of an experienced coach, the players learned a new strategy.