Ytukyg

(The local regularity theorem) Let $P$ be a differential operator of order $k$ that is elliptic over $bar{U}, U subseteq Bbb R^n$ relatively comapct. Let $k,l$ be integers, $f in W^l$ and $u in W^r$. Assume that $Pu=f$, this equation takes place in $W^{r-k}$. Then for each function $mu in C_c^infty(U)$, $mu u in W^{l+k}$.

The spaces where $W^l$ is the completion of the Schwartz space $S(Bbb R^n)$ under Sobolev $l$-norm.
$$ ||f||_l^2 := int (1+|xi|^2)^l |hat{f} (xi)|^2 , dxi $$
For an open set $U subseteq Bbb R^n$, we define $W^l(U)$ to be the completion of $C_c^infty(U)$ under the Sobolev $l$-norm.

What confuses me is the appearance of the integer $r$. Is $r$ supposed to equal to $l$?

Is the statement true even when $k=0$?

Original source maybe hepful: page 46, Theorem 3.5.1.

Depending on how you interpret the expression $Pu$, the prerequisite $uin W^{r}(U)$ can be completely dispensed with. In fact there is the following stronger statement (concerning the regularising properties of an elliptic order $k$ differential operator $P$), which works for $u$ only being a distribution:

If $u in mathscr{D}'(U)$ and $Pu in W^l(U)$, then $mu uin W^{l+k}(U)$ for all $muin C_c^infty(U)$.

Now I suppose that your lecture notes do not assume familiarity with distributions, so we need to make sense of $Pu$ in a more conservative way. Well, if $rge k$ and $uin W^{r}(U)$, then you know what $Pu$ is and the statement reduces to the one you have mentioned.

To illustrate the difference between the two formulations, consider the following example: Let
$$uin L^1_{loc}(U) text{ with }int_U u Delta varphi = 0 text{ for all } varphi in C_c^infty(U). tag{$star$}$$
Now your formulation of elliptic regularity gives the following: If we additionally assume that $u in W^2(U)$, then we can integrate by parts and see that $Delta u = 0$, which lies in $W^l(U)$ for all $lge 0$. Since $Delta$ is elliptic, this implies that $mu uin W^{l+2}(U)$ for all $l$ and thus $u$ is smooth.
However, using the stronger formulation, we do not need to assume that $u$ has any weak derivative: In fact ($star$) is enough to conclude that $u$ is smooth.

Concerning your second question: Yes, it works for $k=0$ and I would say that then the statement is trivial: An order zero $DO$ (with smooth coefficients) has the form $Pu = a u$, where $ain C^infty(U)$ and ellipticity means that $a(x)neq 0$ for all $xin U$. In particular $mu u = (mu a^{-1}) Pu$ and multiplying a function in $W^l(U)$ with $mu a^{-1}in C_c^infty(U)$ certainly yields a function in $W^l(U)$ again.