Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$
At various, places we see that (for instance: in the proof of Hilbert transform is bounded on $1<p<\infty$ ), if the operator is bounded on for the range $1<p<2,$ then the operator is bounded on $L^{p}$ for $p>2$ by duality argument.
My Vague Question is: What is the standard duality argument in these kind of situations? Can you illustrate some specific example?
$\newcommand\ip[2]{\left\langle #1,#2\right\rangle}$
Leaving out many technical details, just illustrating what that "duality argument" is:
Let's define $$\ip fg=\int fg.$$
Suppose that $K$ is a kernel and define $\tilde K(t)=K(-t)$. Then, assuming you can justify the Fubini (typically by restricting to some nice subspace of $L^p$), you see that $$\ip{\tilde K*f}g=\ip f{K*g}.$$
Suppose we've shown that $$||K*f||_p\le c||f||_p.$$ It follows that $||\tilde K*f||_p\le c||f||_p$. So for $f\in L^p$ and $g\in L^{p'}$ we have $$\left|\ip f{K*g}\right|=\left|\ip{\tilde K*f} g\right|\le c||f||_p||g||_{p'},$$which shows that $$||K*g||_{p'}\le c||g||_{p'}.$$