1549055259-Ubiquitous_Quasidisk__The__Gehring_

6.2. HARDY-LITTLEWOOD PROPERTY 79

then f E Lipa(B) with

II! Ila S com

a

where co is an absolute constant.

This result can be extended to uniform domains ([64]) and hence to quasidisks

by Theorem 3.4.5.

THEOREM 6.2.4 (Gehring-Martio [64]). Suppose D is a uniform domain in R^2.

If f is an analytic function in D which satisfies

(6.2.5) lf'(z)I Sm dist(z, &D)°'-^1

for z ED and a E (0, 1], th en

lf(z1) - f(z2)I S cmlz1 - z2I°'

a

for z 1 , z2 E D. Here c is a constant that depends only on the domain D.

PROOF. Fix z1, z2 ED. Since Dis uniform, there is a curve I joining z 1 and

z 2 in D satisfying

length(!) S alz1 - z2I,

min length(tj) Sa dist(z, D), z E /,

J=l,2

where /1, /2 are the components of / \ { z}. Then

lf(zi) ~ f(z2)I S !, lf'(()l ld(I

Sm!, dist((, &D)°'-^1 ld(I

1

length(l')/2 2mal-a (lengt2h(1))°'

< 2ma^1 - a s^1 - °'ds = ---

• o a
  S 21 -°'amlz1 - z2I°' S cmlz1 - z2I°'
  a a
  where c = 2a. 0

DEFINITION 6.2.6. We say that a proper subdomain D of R^2 has the Hardy-

Littlewood property if there exists a constant c = c(D) such that for 0 < a :=:; 1, f

is in Lipa (D) with llflla Sc/a whenever f is analytic with

(6.2.7) lf'(z) I:::; dist(z, oD)°'-^1

in D. Following Ka ufman-Wu [99] we say that D has the Hardy-L ittlewood property
of order a if there is a constant k = k(D, a) such that if f satisfies (6.2.7), then
llflla S k.
REMARK 6.2.8. If D has the Hardy-Littlewood property, then it has the Hardy-
Littlewood property of any order a, while the converse is not true. See [15] for a
counterexample.

By Theorem 6.2.4 all uniform domains and hence all quasidisks have the Hardy-
Littlewood property. Unfortunately this property does not characterize quasidisks
directly, as the following example due to Lappalainen [111] shows. A similar ex-
ample is constructed by Gehring and Martio in [66, 2.26].