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I'm trying to understand a step in a proof. I don't know which of the prerequisites are required or even helpful for what I need, so I give you the complete situation:

Let $T \subset \mathbb{R}^n$ be bounded and convex, let $k\ge 1$ be an integer and let $a\in H^{k,\infty}(T)$. Let $w$ be a continuous function on $T$ (in my situation a polynomial of degree $2k-2$, restricted to $T$). Then, some numeric integration is performed. In particular, the following expression makes my head ache:

$\sum_{l=1}^{L}c_l\cdot(aw)(y_l)$,

where $c_l\ge 0$ and $y_l$ are points in $T$. I.e. the product funciton $aw$ is evaluated at points.

But how is this well defined? I tried to find a continuous representative of $a$ using the Sobolev embedding theorems, but all the versions of them I know of exclude the case where $p=\infty$...

Any hints or thoughts are welcome :) Thank you!

1 Answers 1

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If $T \subset \mathbb{R}^n$ is bounded, and $f \in L^{\infty}(T)$, then $f \in L^p(T)$ for all $1 \leq p < \infty$. This is because $ \int_T |f|^p \leq \int_T ||f||_{\infty}^p = vol(T) ||f||_{\infty}^p < \infty. $

Hence, if $a \in H^{k,\infty}(T)$, then $a$ is also in $H^{k,p}(T)$ for all $1 \leq p < \infty$. It you take $p$ such that $k - \frac{n}{p} > 0$, or $p > \frac{n}{k}$, you obtain from the Sobolev embedding theorems that $a$ has a continuous representative.