Conformally Invariant Variational Problems. - SAM

lemma VII.1 to deduce that for every 0 < δ < 1 there holds
‖∇V‖ 2 L 2,∞ (B δr (p)) ≤ ‖∇V‖ 2 L 2 (B δr (p))
≤
( ) 2 4δ
‖∇V‖ 2 L
3
2 (B 3r/4 (p))
≤ C 1
( 4δ
3
) 2
‖∇V‖ 2 L 2,∞ (B r (p)) ,
(VII.41)
where C 1 is a constant independent of r. Indeed, the L 2,∞ -norm
of a harmonic function on the unit ball controls all its other
norms on balls of radii inferior to 3/4.
We
(
next choose δ independent of r and so small as to have
C 4δ
) 2
1 3 < 1/16. We alsoadjustε0 tosatisfyC 0 ε 0 < 1/8. Then,
combining (VII.40) and (VII.41) yields the following inequality
‖∇W‖ L
2,∞
(B δr (p)) ≤ 1 2 ‖∇W‖ L 2,∞ (B r (p)) ,
(VII.42)
valid for all r < ρ 0 and all p ∈ B 1/2 (0).
Just as in the regularity proof for the CMC equation, the latter
is iterated to eventually produce the estimate
sup ρ −α ‖∇W‖ L
2,∞
(B ρ (p)) < +∞ .
p∈B 1/2 (0) , 0