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 \| 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 \| . 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

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