Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$?
By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ large enough) $u \in W^{k+l,p}$ if $Lu \in W^{k,q}$ and if the coefficients of $L$ are in $W^{k,r}$ (in divergence form, for some $1 < p,q,r < \infty$).
The results in the book of Gilbarg-Trudinger require the coefficients to be in $C^{k,1}$ (by embedding one loses too much to get somewhere). Is $l=1$ possible for a second order operator? Or even $l=2$? If yes, for which $k,p,q,r$?
