I am finishing up a class on function theory and I am trying to reconcile
a few statements that I have seen.
Let us define $L^p(\mathbb R^n)$ to be the set of measurable functions $f$ so that
$\int_{\mathbb R^n} |f|^p dx < \infty$, with the norm $||f||_p = \Big(\int_{\mathbb R^n
} |f|^p dx\Big)^{1/p}$.
I have also seen it written that $L^p(\mathbb R^n)$ is the set of all tempered distributions
$f \in S'$ satisfying $ ||f||_p < \infty$. Here $S$ is the collection of rapidly decreasing complex valued functions on $\mathbb R^n$, and $S'$ is the space of continuous linear functionals on $S$ (tempered distributions).
I would like to show that these two definitions "are the same". However, I am at a disadvantage, since I don't even see how to define the $||\cdot||_p$ norm on
the space of tempered distributions.
Thus, how do we define the $L_p$ norm of a tempered distribution?
Best Answer
Distributions act on the space $\mathcal{D}$ of test functions (infinitely differentiable with compact support). For all $p$ it holds that $\mathcal{D}\subset L^p$ and hence, it makes sense to take the $p$-norm test functions for any $p$. Now take some $p$ and its conjugate $q$ (i.e. $p+q=pq$). For any distribution $T$ we can define $$ \|T\|_p = \sup\{ T(\phi)\ :\ \|\phi\|_q\leq 1\} $$ which may or may not be finite.
Note that this produces the $p$ norm of a function $f$ when applied to the distribution induced by $f$.