The characterization of functions arising as potentials. II

1. Statement of result. We continue our study of the function spaces Z£, begun in [7]. We recall that f^Ll(En) when f=Ka*<l>, where <££!/ (£„) . Ka is the Bessel kernel, characterized by its Fourier transform Ka(x) * (1 +1 x \ )~ a / . I t should also be recalled that the space L?, \<p < <*>, with k a positive integer, coincides with the space of functions which together with their derivatives up to and including order k belong to IS; (see [2]). I t will be convenient to give the functions in La their strict definition. Thus we redefine them to have the value (Ka * cj>)(x) a t every point where this convolution converges absolutely. With this done, and if a—(n — m)/p>0, then the restriction of an fC£L%(En) to a fixed m-dimensional linear variety in En is well-defined (that is, it exists almost everywhere with respect to m-dimensional Euclidean measure). The problem that arises is of characterizing such restrictions. The problem was previously solved in the following cases: (i) When p is arbitrary, but a = 1, in Gagliardo [3]. (ii) When p — 2> and a is otherwise arbitrary in Aronszajn and Smith [ l ] . In each case the solution may be expressed in terms of another function space, Wu, which consists of those ƒ £ ! / ( £ „ ) for which the norm