The harmonic Cesáro and Copson operators on the spaces $L^p({\Bbb R})$, $1\le p \le 2$
Volume 149 / 2002
Abstract
The harmonic Cesàro operator ${\cal C}$ is defined for a function $f$ in $L^p({\mathbb R})$ for some $1\le p <\infty$ by setting ${\cal C}(f) (x) := \int^\infty_x ({f(u)/ u})\, du$ for $x>0$ and ${\cal C}(f)(x) := - \int^x_{-\infty} ({f(u)/ u}) \, du$ for $x<0$; the harmonic Copson operator $\mathbb C^*$ is defined for a function $f$ in $L^1_{{\rm loc}} ({\mathbb R})$ by setting ${\cal C}^*(f) (x) := ({1/ x}) \int^x_0 f(u)\, du$ for $x\not= 0$. The notation indicates that $\mathbb C$ and $\mathbb C^*$ are adjoint operators in a certain sense.
We present rigorous proofs of the following two commuting relations:
(i) If $f\in L^p ({\mathbb R})$ for some $1\le p \le 2$, then $({\cal C}(f))^\wedge (t) = {\cal C}^* (\skew3\widehat{f}\hskip1pt)(t)$ a.e., where $\skew3\widehat{f}$ denotes the Fourier transform of $f$.(ii) If $f\in L^p ({\mathbb R})$ for some $1< p\le 2$, then $({\cal C}^* (f))^\wedge (t)={\cal C} (\skew3\widehat{f}\hskip1pt) (t)$ a.e.
As a by-product of our proofs, we obtain representations of $({\cal C}(f))^\wedge (t)$ and $({\cal C}^*(f))^\wedge (t)$ in terms of Lebesgue integrals in case $f$ belongs to $L^p({\mathbb R})$ for some $1< p\le 2$. These representations are valid for almost every $t$ and may be useful in other contexts.
