ABSTRACT

The spaces H (R ) , 0pl

9

consist of tempered distributions f

for which the maximal function sup |f*+ (x)| belongs to LP(R ) . Here

t0

L

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 ) .