Abstract. We prove a new sufficient condition for a locally absolutely continuous function $F$ that vanishes at infinity to be the Fourier transform of a function $f\in L^1({\msbm R})$ (in sign: $F\in\hat L ({\msbm R}))$ or that of a function $f\in H^1 ({\msbm R})$ (in sign $F\in\hat H ({\msbm R}))$. Namely, if $F'\in\hat H^1({\msbm R})$, then (i) $F\in\hat L({\msbm R})$, (ii) $F\in\hat H({\msbm R})$ if and only if $F(0)=0$. We reformulate two preliminary results, which give also sufficient conditions for $F$ to belong to $\hat L({\msbm R})$ or $\hat H({\msbm R})$, respectively. Finally, we characterize the endomorphisms and the automorphisms of ${\cal M}_1({\msbm R})$, the Banach algebra of all $L^1({\msbm R})$ multipliers.

AMS Subject Classification (1991): 42A38, 43A22, 46J15

Keyword(s): Fourier transform, Hilbert transform, Hardy inequality, multiplier, Fourier-Stieltjes transform, Banach algebra, endomorphism, automorphism

Received July 21, 1994, and in revised form January 12, 1995. (Registered under 5687/2009.)