The characterization of functions arising as potentials. II
The characterization of functions arising as potentials. II
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DOI：
10.1090/s0002-9904-1962-10856-8
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发表时间：
1961
影响因子：
1.3
通讯作者：
BY E. M. Stein;E. Stein
中科院分区：
文献类型：
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作者：
BY E. M. Stein;E. Stein
1.结果陈述。我们继续研究函数空间Ze，开始于[7]。我们回想一下，当f=Ka* 时，f^Ll（En），其中k是正整数，与函数空间一致，这些函数及其导数直到并包括阶k属于IS;（见[2]）。这将是方便的功能在La的严格定义。因此，我们将它们重新定义为在该卷积绝对收敛的每个点处具有值（Ka * cj>）（x）。这样做，如果a-（n-m）/p>0，则fC ε L%（En）对En中的固定m维线性簇的限制是明确定义的（即，它几乎存在于关于m维欧几里得测度的任何地方）。出现的问题是如何描述这种限制。这个问题以前在下列情况下得到了解决：（i）当p是任意的，但a = 1时，在Gagliardo [3]中。(ii)在Aronszajn和Smith [1]中，当p - 2>且a是任意的时，在每种情况下，解可以用另一个函数空间Wu来表示，Wu由那些Wu！/（ε n），其中，
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*, where , 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