Harmonic semifields.

Communicated by S. S. Cherm, October 31, 1966 1. The solution of the principal field problem on arbitrary Riemannian spaces is given in reference 2. In the present note we consider the class S of harmonic semifields, intermediate between the class F of harmonic fields and the class H of harmonic forms. The definitions are as follows. On a given Riemannian space V let A' = 6d, A` = d6, A = A' + A". (1) Then F = {IadC = ba = 01, S = {IatX'Aa A= a = 012 (2) H = {alAa= 01, on V. Clearly F cS c H. Let D be the space of Cdifferential forms with compact supports on V. We form the spaces EA, = A'D, EAU = AID, EAA,, = Ea. + EAf, (3) where the closures are in the space E of square integrable differential forms on V. We also consider the spaces OA'= { alAa= , A ={a "+A+° (4) In contrast with 6 and d, the operator A' is self-adjoint, (Oda, A3) = (a, MdA), and the same is true of A". As a consequence, EA,, 1 Em-, OA' = Ek1, OA = ElA, (5) and we have the following geometric scheme. The 3-space shall stand for the space E. For the x-axis we take EA,, and for the y-axis EAN, so that the xy-plane is EAA,. The xz-plane becomes On, and the yz-plane OA,. The z-axis is the space S of harmonic semifields: A'a = AYa = 0. We know that S c Ca. 2. Let V1 be a regular neighborhood of the ideal boundary of V and let a E sU1) n c.(V). If liall < a, then we take the projection r of a on EA,,,, and the desired principal semifield p = ar can be directly taken from our model. In the general case the solution is as follows: THEOREM. If a ES(E ) fln C(V), then the principal semifield p E S(V) characterized by fp E EA,' exists if and only if