2 BAERNSTEIN AND SAWYER
In §3 we formulate a Fourier multiplier theorem for H which sharpens
up to their natural limits results of Calderon-Torchinsky [CT] and
Taibleson-Weiss. In §4 we prove the embedding theorem and in §5 demonstrate
its sharpness. §6 contains the proof of the multiplier theorem and §7 shows
Finally, in §8-10 we prove three theorems about Fourier transforms of
H distributions which follow easily from the "atomic decomposition". The
first asserts that H has the "lower majorant property" and answers a
question of Weiss. The second contains an H analogue of a recent theorem
of Pigno and Smith, while the last extends a theorem of D.M. Oberlin.
In some respects this paper may be regarded as a successor to [TW], and
we are grateful to Professors Taibleson and Weiss for their friendly interest
and encouragement. We also thank John Fournier for suggesting that we look
in the direction of homogeneous Besov spaces in order to find sharp results.
In [TW] the H distributions are defined as certain linear functionals on
Lipschitz spaces. Latterfs theorem about the atomic decomposition [L], see
[Wi] for another proof and §6 of this paper for a description of the result,
shows that the TW spaces H ^ , where N = [n(— - 1)] , coincide with
the H spaces as defined by us.