In the book, Multiple Integrals in the Calculus of Variations (1996) Morrey defines a cut off function: (middle of the page 35)
$$\eta=\begin{cases} 1, & \text{on } D\subset D^\prime_a,\\ 1-2a^{-1}d(x,D),& \text{ for } 0\le d(x,D)\le a/2,\\ 0, & \text{otherwise.} \end{cases} $$
In page 164 (Lemma 5.7.1) he uses this function and it seems to me that he makes the following estimates:
$$|\nabla \eta|\le C a^{-1},$$
$$|\Delta \eta|\le C a^{-2}.$$
Is that correct or am I misreading?
If yes, can I take third order derivatives of the cut-off function and if the answer is positive to that does it hold that
$$|\nabla^3 \eta|\le C a^{-3}\text{?}$$
Any references for the derivatives of cut off function would be very helpful.