A NULLSTELLENSATZ FOR AMOEBAS

DISTORTION OF HAUSDORFF MEASURES AND REMOVABILITY 545
Lemma 2.1 is a first step toward the results on absolute continuity, as presented in the
following reformulation of Theorem 1.1.
THEOREM 2.2
Let E be a compact set, and let φ : C → C be K-quasiconformal, normalized by
(2.2). Then
M 1( φ(E) ) ≤ C ( M 2/(K+1) (E) ) (K+1)/(2K)
,
where the constant C = C(K) depends only on K. In particular, if H 2/(K+1) (E) = 0,
then H 1( φ(E) ) = 0.
Proof
There is no restriction if we assume that E ⊂ D. We can also assume that φ is a
principal K-quasiconformal mapping, conformal outside D.Now,sinceE is compact,
for any ε>0 there is a finite covering of E by open disks D j , j = 1,...,m, such
that
n∑
j=1
r 2/(K+1)
j ≤ M 2/(K+1) (E) + ε.
By Vitali’s covering lemma, we can replace our covering by a new finite family of
disjoint disks, also denoted D j = D(z j ,r j ), j = 1,...,m, such that E is contained
in the union of 5D j = D(z j , 5r j ). Denote now = ⋃ n
j=1 5D j.Asin[4], we use
a decomposition φ = h ◦ φ 1 , where both φ 1 and h are principal K-quasiconformal
mappings. Moreover, we may require that φ 1 be conformal in ∪ (C \ D) and that h
be conformal outside φ 1 ().
By Lemma 2.1, we see that
M 1( φ(E) ) ≤ M 1( φ() ) = M 1( h ◦ φ 1 () ) ≤ C M 1( φ 1 () ) .
Hence the problem has been reduced to estimating M 1 (φ 1 ()). For this, K-quasidisks
have area comparable to the square of the diameter,
diam ( φ 1 (5D j ) ) ≃ diam ( φ 1 (D j ) ) ( ∫ 1/2
≃|φ 1 (D j )| 1/2 = J (z, φ 1 ) dA(z))
D j
with constants that depend only on K. Thus, using Hölder estimates twice, we obtain
n∑
diam ( φ 1 (5D j ) ) n∑
≃ diam ( φ 1 (D j ) ) ( n∑
∫
≤ C(K) J (z, φ 1 ) p dA(z)
D j
j=1
j=1
j=1
( n∑
) 1−1/(2p),
× |D j | (p−1)/(2p−1)
j=1
) 1/(2p)