DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

INTEGRAL ESTIMATES FOR SOLUTIONS
OF A-HARMONIC TENSORS
BING LIU
We first discuss some properties of a class of A r,λ weighted functions. Then we prove a new
version of weak reverse Hölder inequality, local versions of the Poincaré inequality and
Hardy-Littlewood inequality with A r,λ double weights, and Hardy-Littlewood inequality
with A r,λ double weights on δ-John domain for solutions of the A-harmonic equation.
Copyright © 2006 Bing Liu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
1. Introduction and notation
In his book Bounded Analytic Functions [5], John Garnett showed that the A r condition is
one of the necessary and sufficient conditions for both Hardy-Littlewood maximal operator
and Hilbert transform to be bounded on L r (μ) space. Then Neugebauer introduced
A r,λ condition and discussed its properties, see [6]. Since then, there have been many
studies on inequalities with weighted norms that are related to either A r or A r,λ conditions,
see [1–4]. In this paper we discuss some properties of A r,λ double weights. We
prove a new version of weak reverse Hölder inequality, local versions of the Poincaré inequality
and Hardy-Littlewood inequality with A r,λ -double weights. As an application,
we prove Hardy-Littlewood inequality on δ-JohndomainwiththeA r,λ double weights
for solutions of the A-harmonic equation. First, we introduce some notations.
We denote Ω as a connected open subset of R n . The weighted L p -norm of a measurable
function f over E is defined by
(∫
‖ f ‖ p,E,w =
E
∣ f (x) ∣ 1/p w(x)dx) p . (1.1)
The space of differential l-forms is denoted as D ′ (Ω,∧ l ). We write L p (Ω,∧ l )forthelforms
ω(x) = ∑ I ω I (x)dx I = ∑ ω i1i 2...i l
(x)dx i1 ∧ dx i2 ∧···∧dx il with ω I ∈ L p (Ω,R) for
all ordered l-tuples I.ThenL p (Ω,∧ l ) is a Banach space with norm
(∫
‖ω‖ p,Ω =
Ω
⎛
∣ ω(x) ∣ 1/p ∫ ( ∑ dx) p = ⎝ ∣ ω I (x) ∣ ) p/2
⎞
2 dx⎠
Ω
I
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 685–697
1/p
. (1.2)