What's the meaning of "on notice" and "under advisement"?

Can someone please give a clear definition and distinction of these terms, as when a public figure is asked a difficult question and says: "I'll take that on notice" or "I'll take that under advisement."

Solution 1:

Let me answer your second question first

take something under advisement reserve judgment while considering something.

It's usually used by a judge when he announces he is going to consider what counsel has said and rule on it later.

As for "I'll take that on notice," I haven't heard that used. Usually the term is to "put someone on notice" meaning to alert someone to a fact and to the fact that you have given them some kind of ultimatum:

The police put Niko Bellic on notice that no further racketeering would be tolerated in Liberty City.

It's a way of telling someone "You have been warned."