Example of an unbounded operator

Question

Can somebody give me an easy example of a linear operator which maps $L^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$ and $L^\infty(\mathbb{R}^n)$ to $L^\infty(\mathbb{R}^n)$ (but not boundedly) but does not admit a bounded extension from $L^2(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$ (or any other $L^p$, $1<p<\infty$) ?

Answer 1

There are no explicit (easy or otherwise) examples of unbounded linear operators (or functionals) defined on a Banach space. Their very existence depends on the axiom of choice. See Discontinuous linear functional.