Hubert Transforms, Fractional Integration and Differentiation by P. L. Butzer and W. Trebels

The purpose of this note is to announce a number of results concerning the various approaches to fractional integration on the real line E as due to H. Weyl [9], M. Riesz [7], W. Feller [4] and G. O. Okikiolu [ô]. Our principal contributions are on extensions of theorems of J. L. B. Cooper [3], on the interchange of the operations of fractional integration (differentiation) and the Hilbert transform, on the counterpart of a theorem of H. Weyl [9] on periodic functions to the real line, and on partial differential equations of fractional order. Since Fourier transform methods are mainly used in the proofs, the discussion is restricted to the space L(E) for l^g£(£) , 1 £p g 2. We define the Hilbert transform of ƒ by