NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

116 GUILLARMOU and TZOU
Therefore, by the Riesz theorem, there exists vf ∈ A such that
(g + V )vf (g + Vi)w dvg = wf dvg
M0
for all w ∈ B. Furthermore, (g + Vi)vf L2 ≤fB∗. Setting u := (g + Vi)vf ,
we have (g + Vi)u = f and uL2 ≤fB∗. To obtain the boundary condition for
u, observe that since
M0
u(g + Vi)wdvg =
for all w ∈ B, then by Green’s theorem,
u∂νw dvg = 0 =
M0
Ɣ0
M0
M0
fwdvg
u∂νw dvg
for all w ∈ B. Thisimpliesthatu = 0 on Ɣ0.
Clearly, it suffices to assume that Ɣ is a small piece of the boundary which is contained
in a single coordinate chart with complex coordinates z = x + iy, where |z| ≤1,
Im(z) > 0, and the boundary is given by {y = 0}. Moreover, the metric is of the
form e 2ρ |dz| 2 for some smooth function ρ.Letp ∈ Ɣ, and possibly by translating the
coordinates, we can assume that p ={z = 0}. Letη ∈ C ∞ (M0) be a cutoff function
supported in a small neighborhood of p. Forh>0 small, we define the smooth
function vh ∈ C ∞ (M0) supported near p via the coordinate chart Z = (x,y) ∈ R 2 by
vh(Z) := η(Z/ √ h)e (1/h)α·Z , (A.2)
where α := (i, −1) ∈ C 2 is chosen such that α · α = 0 and the dot product · means
the canonical symmetric C-bilinear form on C 2 . Thus we get (∂ 2 x + ∂2 y )eα·Z = 0 and
thus ge α·Z = 0 by conformal covariance of the Laplacian. Therefore, we have in
local coordinates
gvh(Z) = 1
h e(1/h)α·Z (gη)
Z√h
+ 2
Z√h
e(1/h)α·Z dη ,α· dZ . (A.3)
h3/2 g
LEMMA A.3
If V ∈ C 0,α (M0) for q>2, then there exists a solution uh ∈ H 2 to (g + V )u = 0
of the form
uh = vh + Rh,
with vh defined in (A.2) and RhL 2 ≤ Ch5/4 , satisfying supp(Rh|∂M0) ⊂ Ɣ.