DOI: 10.68381/jca22027 ISSN: 0944-6532
Boundedness Criterions for the Hardy Operator in Weighted L
p(·)
(0,l) Space
Farman Mamedov, Firana M. Mammadova, Mushviq Aliyev
Equivalent conditions are proved for the Hardy type weighted inequality
\Big\Vert W(\cdot)^{-1}\sigma(\cdot)^{\frac{1}{p(\cdot)}} \int_{0}^{x} f(t)dt \Big \Vert_{L^{p(\cdot)}(0,l)} \leq C \Big \Vert \omega(\cdot)^{ \frac{1}{p(\cdot)}} f(\cdot) \Big \Vert_{L^{p(\cdot)}(0,l)}, \; \; \; f \geq 0
∥
W
(
⋅
)
−
1
σ
(
⋅
)
1
p
(
⋅
)
∫
0
x
f
(
t
)
d
t
∥
L
p
(
⋅
)
(
0
,
l
)
≤
C
∥
ω
(
⋅
)
1
p
(
⋅
)
f
(
⋅
)
∥
L
p
(
⋅
)
(
0
,
l
)
,
f
≥
0
to be fulfilled in the norms of a Lebesgue space with variable exponent
L^{p(\cdot)}(0,l)
L
p
(
⋅
)
(
0
,
l
)
. It is assumed that the function
p(.)
p
(
.
)
is a monotone function.