Theorem 2.3.8 in 'The Geometry of Four-Manifolds' by Donaldson and Kronheimer reads:
There is a constant $\epsilon>0$ such that if $A$ is a ASD (anti self dual) connection on the trivial bundle over $B^4$ which satisfies the Coulomb gauge condition $d^* A =0$ and $||A||_{L^4} \leq \epsilon$, then for any interior domain $D \Subset B^4$ and $l \geq 1$ we have $$||A||_{L^2_l(D)} \leq M_{l,D} ||F_A||_{L^2(B^4)} $$
This is a result that is supplementary to Uhlenbeck's theorem.
The strategy of proof in the book is as follows: Lemma 2.3.10 asserts the inequality for connections over $S^4$ satisfying $d^* A=0$ and for $l=1$. (The proof of) Lemma 2.3.11 bounds the $L^2_{l+1}$ norm of the connection in terms of the $L^2_{l}$ norm, so that effectively we now have the case for $S^4$ and $d^*A=0$, and for any $l$.
However the last step perplexes me, given a connection $A$ over $B^4$, the authors want to consider $B^4 \subset S^4$, then extend $A$ to $S^4$ to use the lemmas stated above. The way they extend the connection is simply multiplying by a cut-off function. But to use the lemmas, we need the Coulomb gauge condition $d^* A=0$, which is not preserved after multiplying by a cut-off function, so this approach is not valid!
One might wonder if the lemmas apply to $B^4$ as well. However, I cannot generalize the proofs for this to work, for it requires elliptic estimates, which do not hold on non-compact manifolds (at least not without some additional hypothesis). Any help is appreciated!