Abstract
In this article, we introduce anisotropic mixed-norm Herz spaces
K
˙
q
→
,
a
→
α
,
p
(
R
n
)
{\dot{K}}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n})
and
K
q
→
,
a
→
α
,
p
(
R
n
)
{K}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n})
and investigate some basic properties of those spaces. Furthermore, we establish the Rubio de Francia extrapolation theory, which resolves the boundedness problems of Calderón-Zygmund operators and fractional integral operator and their commutators, on spaces
K
˙
q
→
,
a
→
α
,
p
(
R
n
)
{\dot{K}}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n})
and
K
q
→
,
a
→
α
,
p
(
R
n
)
{K}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n})
. Especially, the Littlewood-Paley characterizations of anisotropic mixed-norm Herz spaces are also gained. As the generalization of anisotropic mixed-norm Herz spaces, we introduce anisotropic mixed-norm Herz-Hardy spaces
H
K
˙
q
→
,
a
→
α
,
p
(
R
n
)
H{\dot{K}}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n})
and
H
K
q
→
,
a
→
α
,
p
(
R
n
)
H{K}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n})
, on which atomic decomposition and molecular decomposition are obtained. Moreover, we gain the boundedness of classical Calderón-Zygmund operators.