For each () it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to . There is an exchange theorem and inversion in norm.
The harmonic Cesàro operator is defined for a function f in for some 1 ≤ p < ∞ by setting for x > 0 and for x < 0; the harmonic Copson operator ℂ* is defined for a function f in by setting for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If for some 1 ≤ p ≤ 2, then a.e., where f̂ denotes the Fourier transform of f. (ii) If for some 1 < p ≤ 2, then a.e. As...
It is shown that the Muckenhoupt structure constants for f and f* on the real line are the same.