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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 &gt; ∞  \smallint {\;_{{R^n}}}|\hat f(x)|m(|x|)\;dx &gt; \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 : −  ∞  &gt; k &gt; ∞  } &gt; ∞  \sup \{ {2^{(n - 1)k}}\smallint _{{2^k}}^{{2^{k + 1}}}m(r)\;dr: - \infty &gt; k &gt; \infty \} &gt; \infty      .

A multiplier theorem for $H\sp{1}({\bf R}\sp{n})$