We prove the following theorem. Theorem. Let m be a nonnegative measurable function on
[
0
,
∞
)
[0,\infty )
. For
n
⩾
2
n \geqslant 2
, the two conditions below are equivalent: (a)
∫
R
n
|
f
^
(
x
)
|
m
(
|
x
|
)
d
x
>
∞
\smallint {\;_{{R^n}}}|\hat f(x)|m(|x|)\;dx > \infty
for each
f
∈
H
1
(
R
n
)
f \in {H^1}({R^n})
, (b)
sup
{
2
(
n
−
1
)
k
∫
2
k
2
k
+
1
m
(
r
)
d
r
:
−
∞
>
k
>
∞
}
>
∞
\sup \{ {2^{(n - 1)k}}\smallint _{{2^k}}^{{2^{k + 1}}}m(r)\;dr: - \infty > k > \infty \} > \infty
.