If $$\psi :[0,\ell ]\rightarrow [0,\infty [$$
ψ
:
[
0
,
ℓ
]
→
[
0
,
∞
[
is absolutely continuous, nondecreasing, and such that $$\psi (\ell )>\psi (0)$$
ψ
(
ℓ
)
>
ψ
(
0
)
, $$\psi (t)>0$$
ψ
(
t
)
>
0
for $$t>0$$
t
>
0
, then for $$f\in L^1(0,\ell )$$
f
∈
L
1
(
0
,
ℓ
)
, we have $$\begin{aligned} \Vert f\Vert _{1,\psi ,(0,\ell )}:=\int \limits _0^\ell \frac{\psi '(t)}{\psi (t)^2}\int \limits _0^tf^*(s)\psi (s)dsdt\approx \int \limits _0^\ell |f(x)|dx=:\Vert f\Vert _{L^1(0,\ell )},\quad (*) \end{aligned}$$
‖
f
‖
1
,
ψ
,
(
0
,
ℓ
)
:
=
∫
0
ℓ
ψ
′
(
t
)
ψ
(
t
)
2
∫
0
t
f
∗
(
s
)
ψ
(
s
)
d
s
d
t
≈
∫
0
ℓ
|
f
(
x
)
|
d
x
=
:
‖
f
‖
L
1
(
0
,
ℓ
)
,
(
∗
)
where the constant in $$ > rsim $$
≳
depends on $$\psi $$
ψ
and $$\ell $$
ℓ
. Here by $$f^*$$
f
∗
we denote the decreasing rearrangement of f. When applied with f replaced by $$|f|^p$$
|
f
|
p
, $$1<p<\infty $$
1
<
p
<
∞
, there exist functions $$\psi $$
ψ
so that the inequality $$\Vert |f|^p\Vert _{1,\psi ,(0,\ell )}\le \Vert |f|^p\Vert _{L^1(0,\ell )}$$
‖
|
f
|
p
‖
1
,
ψ
,
(
0
,
ℓ
)
≤
‖
|
f
|
p
‖
L
1
(
0
,
ℓ
)
is not rougher than the classical one-dimensional integral Hardy inequality over bounded intervals $$(0,\ell )$$
(
0
,
ℓ
)
. We make an analysis on the validity of $$(*)$$
(
∗
)
under much weaker assumptions on the regularity of $$\psi $$
ψ
, and we get a version of Hardy’s inequality which generalizes and/or improves existing results.