INTRODUCTION

The space H

P

(R

n

) ,, 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*^

t

) ( x ) | , x G Rn ,

t 0

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,

p

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.

1