Affiliation:
1. Institute for Advanced Study , Fuld Hall 412, 1 Einstein Drive , Princeton , NJ , United States of America
2. Okinawa Institute of Science and Technology Graduate University, Nonlinear Analysis Unit , 1919–1 Tancha, Onna-son, Kunigami-gun , Okinawa , Japan
Abstract
Abstract
It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality
H
∞
β
(
{
x
∈
Ω
:
|
I
α
f
(
x
)
|
>
t
}
)
≤
C
e
−
c
t
q
′
$$\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}$$
for all
∥
f
∥
L
N
/
α
,
q
(
Ω
)
≤
1
$\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$
and any
β
∈
(
0
,
N
]
,
where
Ω
⊂
R
N
,
H
∞
β
$\beta \in (0,N], \; {\text{where}} \; \Omega \subset \mathbb{R}^N, \mathcal{H}^{\beta}_{\infty}$
is the Hausdorff content, LN
/α,q
(Ω) is a Lorentz space with q ∈ (1,∞], q' = q/(q − 1) is the Hölder conjugate to q, and Iαf denotes the Riesz potential of f of order α ∈ (0, N).
Cited by
6 articles.
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