The spaces H (R ) , 0pl
consist of tempered distributions f
for which the maximal function sup |f*+ (x)| belongs to LP(R ) . Here
i| r C with J \j r = 1 . We prove two main theorems. The first gives sharp
conditions on the "size" of f which imply that f belongs to H . The
conditions are phrased in terms of certain spaces K introduced by Herz. Our
theorem may be regarded as the limiting endpoint version of a theorem by
Taibleson and Weiss involving "molecules". We then use this embedding
theorem to prove a sharp Fourier embedding theorem of Bernstein-Taibleson-
Herz type.
Our other main theorem gives sharp sufficient conditions on m £ L (R ) ,
for m to be a Fourier multiplier of H , This theorem also involves the
K spaces and may be regarded as the limiting endpoint version of a multi-
plier theorem of Calderon and Torchinsky.
We also prove three results about Fourier transforms of H distribu-
tions. The first establishes the "lower majorant property" for H and the
second is an H (R ) version of a recent theorem of Pigno and Smith about
H (TT) . The third result generalizes a theorem of Oberlin about growth of
* I n
spherical means of f , f H (F ) .
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