Affiliation:
1. Department of Applied Mathematics , National Chiao Tung University , 1001 Ta Hsueh Rd, 30010 Hsinchu , Taiwan
2. Department of Mathematics , National Taiwan Normal University , No. 88, Section 4, Tingzhou Road, Wenshan District , Taipei City , Taiwan 116
Abstract
Abstract
In this paper, we define a notion of β-dimensional mean oscillation of functions
u
:
Q
0
⊂
ℝ
d
→
ℝ
{u:Q_{0}\subset\mathbb{R}^{d}\to\mathbb{R}}
which are integrable on β-dimensional subsets of the cube
Q
0
{Q_{0}}
:
∥
u
∥
BMO
β
(
Q
0
)
:=
sup
Q
⊂
Q
0
inf
c
∈
ℝ
1
l
(
Q
)
β
∫
Q
|
u
-
c
|
𝑑
ℋ
∞
β
,
\displaystyle\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}\vcentcolon=\sup_{Q\subset Q_{%
0}}\inf_{c\in\mathbb{R}}\frac{1}{l(Q)^{\beta}}\int_{Q}|u-c|\,d\mathcal{H}^{%
\beta}_{\infty},
where the supremum is taken over all finite subcubes Q parallel to
Q
0
{Q_{0}}
,
l
(
Q
)
{l(Q)}
is the length of the side of the cube Q, and
ℋ
∞
β
{\mathcal{H}^{\beta}_{\infty}}
is the Hausdorff content. In the case
β
=
d
{\beta=d}
we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every
β
∈
(
0
,
d
]
{\beta\in(0,d]}
one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants
c
,
C
>
0
{c,C>0}
such that
ℋ
∞
β
(
{
x
∈
Q
:
|
u
(
x
)
-
c
Q
|
>
t
}
)
≤
C
l
(
Q
)
β
exp
(
-
c
t
∥
u
∥
BMO
β
(
Q
0
)
)
\displaystyle\mathcal{H}^{\beta}_{\infty}(\{x\in Q:|u(x)-c_{Q}|>t\})\leq Cl(Q)%
^{\beta}\exp\biggl{(}-\frac{ct}{\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}}\biggr{)}
for every
t
>
0
{t>0}
,
u
∈
BMO
β
(
Q
0
)
{u\in\mathrm{BMO}^{\beta}(Q_{0})}
,
Q
⊂
Q
0
{Q\subset Q_{0}}
, and suitable
c
Q
∈
ℝ
{c_{Q}\in\mathbb{R}}
. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.
Subject
Applied Mathematics,Analysis
Cited by
1 articles.
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