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

its sharpness.

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.