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Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications
Authors
Dachun Yang
Wen Yuan
Chenfeng Zhu
Publication date
19 July 2023
Publisher
View
on
arXiv
Abstract
Let
Ξ³
β
R
β
{
0
}
\gamma\in\mathbb{R}\setminus\{0\}
Ξ³
β
R
β
{
0
}
and
X
(
R
n
)
X(\mathbb{R}^n)
X
(
R
n
)
be a ball Banach function space satisfying some mild assumptions. Assume that
Ξ©
=
R
n
\Omega=\mathbb{R}^n
Ξ©
=
R
n
or
Ξ©
β
R
n
\Omega\subset\mathbb{R}^n
Ξ©
β
R
n
is an
(
Ξ΅
,
β
)
(\varepsilon,\infty)
(
Ξ΅
,
β
)
-domain for some
Ξ΅
β
(
0
,
1
]
\varepsilon\in(0,1]
Ξ΅
β
(
0
,
1
]
. In this article, the authors prove that a function
f
f
f
belongs to the homogeneous ball Banach Sobolev space
W
Λ
1
,
X
(
Ξ©
)
\dot{W}^{1,X}(\Omega)
W
Λ
1
,
X
(
Ξ©
)
if and only if
f
β
L
l
o
c
1
(
Ξ©
)
f\in L_{\mathrm{loc}}^1(\Omega)
f
β
L
loc
1
β
(
Ξ©
)
and
sup
β‘
Ξ»
β
(
0
,
β
)
Ξ»
β₯
[
β«
{
y
β
Ξ©
:
Β
β£
f
(
β
)
β
f
(
y
)
β£
>
Ξ»
β£
β
β
y
β£
1
+
Ξ³
p
}
β£
β
β
y
β£
Ξ³
β
n
β
d
y
]
1
p
β₯
X
(
Ξ©
)
<
β
,
\sup_{\lambda\in(0,\infty)}\lambda \left\|\left[\int_{\{y\in\Omega:\ |f(\cdot)-f(y)|>\lambda|\cdot-y|^{1+\frac{\gamma}{p}}\}} \left|\cdot-y\right|^{\gamma-n}\,dy \right]^\frac{1}{p}\right\|_{X(\Omega)}<\infty,
Ξ»
β
(
0
,
β
)
sup
β
Ξ»
β
[
β«
{
y
β
Ξ©
:
Β
β£
f
(
β
)
β
f
(
y
)
β£
>
Ξ»
β£
β β
y
β£
1
+
p
Ξ³
β
}
β
β£
β
β
y
β£
Ξ³
β
n
d
y
]
p
1
β
β
X
(
Ξ©
)
β
<
β
,
where
p
β
[
1
,
β
)
p\in[1,\infty)
p
β
[
1
,
β
)
is related to
X
(
R
n
)
X(\mathbb{R}^n)
X
(
R
n
)
. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey, Bourgain--Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice) Sobolev spaces, which is new even in all these special cases; in particular, it is still new even when
X
(
Ξ©
)
:
=
L
q
(
R
n
)
X(\Omega):=L^q(\mathbb{R}^n)
X
(
Ξ©
)
:=
L
q
(
R
n
)
with
1
β€
p
<
q
<
β
1\leq p<q<\infty
1
β€
p
<
q
<
β
. The novelty of this article exists in that, to establish the characterization of
W
Λ
1
,
X
(
Ξ©
)
\dot{W}^{1,X}(\Omega)
W
Λ
1
,
X
(
Ξ©
)
, the authors provide a machinery via using a generalized Brezis--Seeger--Van Schaftingen--Yung formula on
X
(
R
n
)
X(\mathbb{R}^n)
X
(
R
n
)
, an extension theorem on
W
Λ
1
,
X
(
Ξ©
)
\dot{W}^{1,X}(\Omega)
W
Λ
1
,
X
(
Ξ©
)
, a Bourgain--Brezis--Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space
W
1
,
X
(
Ξ©
)
W^{1,X}(\Omega)
W
1
,
X
(
Ξ©
)
, and a method of extrapolation to overcome those difficulties caused by that
X
(
R
n
)
X(\mathbb{R}^n)
X
(
R
n
)
might be neither the rotation invariance nor the translation invariance and that the norm of
X
(
R
n
)
X(\mathbb{R}^n)
X
(
R
n
)
has no explicit expression.Comment: arXiv admin note: text overlap with arXiv:2304.0094
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oai:arXiv.org:2307.10528
Last time updated on 26/07/2023