What is a duality argument?

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I believe this is probably a question that can have a wide variety of answers, but i believe i'm still interested. The thing is, i have been in a couple pde talks, and i saw that in both of these talks they made some estimates calculus refering to duality arguments, and i really don't know what this duality arguments are.

I know that it depends on some cases i can have different duality arguments, but can u give me some examples of some duality arguments, or how this came up on theory.

In summary i would like to have as much information i can about this topic since i really dont know what those duality arguments are, so references would be great.

Thank you in advance.