Abstract
In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator (
B
B
-maximal operator) on
L
p
(
⋅
)
,
γ
(
R
k
,
+
n
)
{L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n})
variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the
B
B
-maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the
B
B
-maximal operator is not bounded on
L
p
(
⋅
)
,
γ
(
R
k
,
+
n
)
{L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n})
variable exponent Lebesgue spaces in the case of
p
−
=
1
{p}_{-}=1
. We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional
B
B
-maximal function) on
L
p
(
⋅
)
,
γ
(
R
k
,
+
n
)
{L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n})
variable exponent Lebesgue spaces.
Commutators of the fractional maximal function on variable exponent Lebesgue spaces
