It is well known that the finite HILBERT transform T is a NOETHER (FREDHOLM) operator when considered as a map from Y p into itself if 1 < p < 2 or 2 < p c co. When p = 2, the map T is not a NOETHER operator. We present two theorems which characterize the range of T in Y 2 and, as immediate consequences, give simple expressions for its inverse.
IntroductionWe consider the finite HILBERT transform T over the open arc 1-1, l[. A well known result of M. RIESZ [12] tells us that the restriction Tp of T to the BANACH space Y p := YPa-1, 10 defines a continuous linear operator from Y p into Y p for every p €11, a[. Unless p = 2, the map T, is a NOETHER operator (for the definition of NOETHER operators, see section 2). The purpose of this paper is to investigate the operator T, : Y z + Y 2 whose range is a proper dense subspace of Y 2 . This we do in sections 3 and 4, after summarizing some basic results in section 2.Our principal result which characterizes the range of T, is given in Theorem 3.2. This is then followed by a simple inversion formula for T2 which is valid for all functions in the range of T2. These results are derived essentially by considering those operators T,, 1 < p < 2, whose restrictions to Y p are 7''.In section4, on the other hand, the properties of the operators T, , 2 < p c 00, are applied to the study of T2. In Theorem 4.2, the subspace of the range of T2, which consists of all functions f such that (1 -x2)-lI2 f is LEBESGUE integrable, is characterized and the restriction of T;' to that subspace is shown to have the same form as T i ' , 2 c p < m.From Theorem 4.2, the question naturally arises as to whether or not there exist functions 4 E Y 2 for which (1 -x2)-'"
