I'm not sure I quite appreciate exactly what is being asked. You lay out two cases and then say that what is essentially just a third case is an example of the other two not working. You say that the Sobolev embedding theorem "fails" or "goes wrong" when $k=n/p$, but one might say that it is simply neither of the two cases you lay out at the start. Nothing "fails", it just happens to be its own special case.

Sorry - so do you know how the osculating space is then defined using this polynomial map?

To clarify, the rest of Tilli's paper - i.e. the non-so-new stuff - looks, at a glance, to be the same as the old argument that's in Han and Lin. So it seems a perfectly good source.

The new point about this paper Mircea seems to be that they avoid the iteration altogether, by basically differentiating' the quantity which is usually treated discretely and iterated. A proof along the lines which Connor describes in his initial answer, i.e. for pure divergence form equations, using iteration but not John-Nirenberg existed right at the start' as it were - it is due essentially to De Giorgi himself. You can find it the book of Han and Lin, Elliptic PDE. The John-Nirenberg Lemma was used by Moser to prove his Harnack inequality, which itself was not crucial for regularity.

Did you find what you wanted?