Abstract
The aim of this paper is to establish the boundedness of commutator
[
b
,
g
˙
r
]
\left[b,{\dot{g}}_{r}]
generated by Littlewood-Paley
g
g
-functions
g
˙
r
{\dot{g}}_{r}
and
b
∈
RBMO
(
μ
)
b\in {\rm{RBMO}}\left(\mu )
on non-homogeneous metric measure space. Under assumption that
λ
\lambda
satisfies
ε
\varepsilon
-weak reverse doubling condition, the author proves that
[
b
,
g
˙
r
]
\left[b,{\dot{g}}_{r}]
is bounded from Lebesgue spaces
L
p
(
μ
)
{L}^{p}\left(\mu )
into Lebesgue spaces
L
p
(
μ
)
{L}^{p}\left(\mu )
for
p
∈
(
1
,
∞
)
p\in \left(1,\infty )
and also bounded from spaces
L
1
(
μ
)
{L}^{1}\left(\mu )
into spaces
L
1
,
∞
(
μ
)
{L}^{1,\infty }\left(\mu )
. Furthermore, the boundedness of [
b
,
g
˙
r
b,{\dot{g}}_{r}
] on Morrey space
M
q
p
(
μ
)
{M}_{q}^{p}\left(\mu )
and on generalized Morrey
L
p
,
ϕ
(
μ
)
{L}^{p,\phi }\left(\mu )
is obtained.
The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space
