The harmonic Cesàro and Copson operators on the spaces \(L^p({\mathbb R})\), \(1\leq p \leq 2\) (Q2773423)

For the harmonic Cesàro operator \(C(f)(x)\) defined for a function \(f\) in \(L^p(\mathbb{R})\), where \(1 \leq p < \infty\) and \(x >0\), and the harmonic Copson operator \(C^*(f)(x)\) defined for a function \(f\) in \(L^1_{\text{loc}}(\mathbb{R})\) and \(x \neq 0\), the author proves the following two commuting relations: NEWLINENEWLINENEWLINE(i) If \(f \in L^p(\mathbb{R})\) for some \(1 \leq p \leq 2\), then \((C(f))^{\wedge}(t)=C^*(\hat f)(t)\) a.e. (ii) If \(f \in L^p(\mathbb{R})\) for some \(1< p \leq 2\), then \((C^*(f))^{\wedge}(t)=C(\hat f)(t)\) a.e., where \(\hat f\) denotes the Fourier transform of \(f\). Moreover, the author obtains representations of \((C(f))^{\wedge}(t)\) and \((C^*(f))^{\wedge}(t)\) in terms of Lebesgue integrals in case \(f\) belongs to \(L^p(\mathbb{R})\) for some \(1 <p \leq 2\). \textit{R. Bellman} [Bull. Am. Math. Soc. 50, 741-744 (1994; Zbl 0060.18404)] with heuristic motivations formulated the results and \textit{B. I. Golubov} [Russ. Acad. Sci., Sb., Math. 83, 321-330 (1995; Zbl 0842.42003)] without recognizing the forms of the Cesàro and Copson operators presented proofs for the results in the case of the cosine Fourier transform.