The space of the multipliers on a Banach function space S is denoted by MS. Let , , , be the space of Bessel potentials on , defined as the completion of by the norm . According to E. M. Stein, let us define the extension (,y) on of a function (x) on by where the kernel satisfies
(*) , , for some and
In this paper it is proved that
(i) if and it follows that with , where , , is the trace of on
(ii) if satisfies (*) and (**) with and if (resp. and , then it follows that with (resp. (Theorem 1, 2).
The proofs are considerably hard partly because some of the notations adopted there are not familiar to the present reviewer.