I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely on the John-Nirenberg lemma. 


I was wondering if somebody can point out a reference for that proof, or a reason why the John-Nirenberg lemma cannot be truly avoided (or both!).



Moser regularity proof avoiding John-Nirenberg lemma
Check this paper
Moser. On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math (1971) vol. 24 (5) pp. 727-740


The purpose of the above paper, is to avoid the use of the parabolic John-Nirenberg lemma.


