The space H
) ,, n l , 0 p 1 , c o n s i s t s of tempered d i s t r i b u t i o n s
lx fo r which th e maximal f u n c t i o n
f*(x ) = max |(f*^
) ( x ) | , x G Rn ,
belongs t o L (R ) . Here \| f i s any f u n c t i o n i n Cn wit h J* | dx = 1 ,
and \j/.(x) = t \jf(t x) . We d e f i n e
| f j | P = P ( f * ) P dx
Many characterizations of H (R ) are given in [FS] . In particular,
it is proved there that H is independent of the choice of \| f .
If f is a function on R which defines a tempered distribution one
can ask what sorts of restrictions on the "size" of f will imply that
f £ H . Taibleson and Weiss [TW] have found one such set of conditions.
They call their functions "molecules". In §1 we present an embedding theorem
of this sort which includes the Taibleson-Weiss results and is "sharp" in
several respects. This enables us in §2 to prove sharp theorems of the form
5 : X - H , where r& denotes Fourier transformation. These results are
H analogues of theorems of S. Bernstein, Taibleson and Herz for L .
Received by the editors February 1, 1984.
The first author was supported by a grant from the National Science Founda-
tion and the second author by a grant from the National Science and
Engineering Research Council of Canada.