There is no phrasal verb 'give down' in English. Perhaps the writer meant to write/say: I won't give UP on you! (which means 'I won't stop trying for you or believing in you') I won't GET down on you (which means they won't do or say bad things about you) I won't TURN you down (which means 'I won't say no to you') I won't LOOK down on you (which means 'I won't think or act like you are lower than anyone else) And there are other phrasal verbs using the preposition down, so I think the writer DID NOT use good English. Hope this helps...

I honestly never heard the expression give down. I have heard hand down. Do you think that is what you mean? Do you have a sentence where you saw it?

What is the sentence in which it appears?

As Judy said, what is the context?