For
1
⩽
p
⩽
∞
1 \leqslant p \leqslant \infty
, we consider
p
p
-integrable functions on a finite cube
Q
0
{Q_0}
in
R
n
{{\mathbf {R}}^n}
, satisfying
\[
(
1
|
Q
|
∫
Q
|
f
(
x
)
−
f
Q
|
p
d
x
)
1
/
p
⩽
C
φ
(
|
Q
|
)
{\left ( {\frac {1} {{|Q|}}\int _Q {|f(x) - {f_Q}{|^p}dx} } \right )^{1/p}} \leqslant C\varphi (|Q|)
\]
for every parallel subcube
Q
Q
of
Q
0
{Q_0}
, where
|
Q
|
|Q|
denotes the volume of
Q
Q
,
f
Q
{f_Q}
is the mean value of
f
f
over
Q
Q
and
φ
(
t
)
\varphi (t)
is a nonnegative function defined in
(
0
,
∞
)
(0,\infty )
, such that
φ
(
t
)
\varphi (t)
is nonincreasing near zero,
φ
(
t
)
→
∞
\varphi (t) \to \infty
as
t
→
0
t \to 0
, and
t
φ
p
(
t
)
t{\varphi ^p}(t)
is nondecreasing near zero. The constant
C
C
does not depend on
Q
Q
. Let
g
g
be a nonnegative
p
p
-integrable function
g
:
(
0
,
1
)
→
R
g:(0,1) \to {\mathbf {R}}
such that
g
g
is nonincreasing and
g
(
t
)
→
∞
g(t) \to \infty
as
t
→
0
t \to 0
. We prove here that there exist a cube
Q
0
{Q_0}
and a function
f
f
satisfying condition
(
1
)
(1)
for every parallel subcube
Q
Q
of
Q
0
{Q_0}
, such that
δ
f
(
λ
)
⩾
C
1
δ
g
(
λ
)
{\delta _f}(\lambda ) \geqslant {C_1}{\delta _g}(\lambda )
for
λ
⩾
λ
0
\lambda \geqslant {\lambda _0}
,
C
1
>
0
{C_1} > 0
, where
δ
(
λ
)
\delta (\lambda )
denotes the distribution function.