The harmonic Cesáro and Copson operators on the spaces $L^p\left(ℝ\right)$, 1 ≤ p ≤ 2

Ferenc Móricz. "The harmonic Cesáro and Copson operators on the spaces $L^{p}(ℝ)$, 1 ≤ p ≤ 2." Studia Mathematica 149.3 (2002): 267-279. <http://eudml.org/doc/285082>.

@article{FerencMóricz2002,
abstract = {The harmonic Cesàro operator is defined for a function f in $L^\{p\}(ℝ)$ for some 1 ≤ p < ∞ by setting $(f)(x): = ∫^\{∞\}_\{x\} (f(u)/u)du$ for x > 0 and $(f)(x): = -∫_\{-∞\}^\{x\} (f(u)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L¹_\{loc\}(ℝ)$ by setting $*(f)(x): = (1/x) ∫^\{x₀\} f(u)du$ 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 $f ∈ L^\{p\}(ℝ)$ for some 1 ≤ p ≤ 2, then $((f))^\{∧\}(t) = *(f̂)(t)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f ∈ L^\{p\}(ℝ)$ for some 1 < p ≤ 2, then $(*(f))^\{∧\}(t) = (f̂)(t)$ a.e. As a by-product of our proofs, we obtain representations of $((f))^\{∧\}(t)$ and $(*(f))^\{∧\}(t)$ in terms of Lebesgue integrals in case f belongs to $L^\{p\}(ℝ)$ for some 1 < p ≤ 2. These representations are valid for almost every t and may be useful in other contexts.},
author = {Ferenc Móricz},
journal = {Studia Mathematica},
keywords = {harmonic Cesàro and Copson operators; Fourier transform; commuting relations; Hardy's inequalities; representations for Fourier transform},
language = {eng},
number = {3},
pages = {267-279},
title = {The harmonic Cesáro and Copson operators on the spaces $L^\{p\}(ℝ)$, 1 ≤ p ≤ 2},
url = {http://eudml.org/doc/285082},
volume = {149},
year = {2002},
}

TY - JOUR
AU - Ferenc Móricz
TI - The harmonic Cesáro and Copson operators on the spaces $L^{p}(ℝ)$, 1 ≤ p ≤ 2
JO - Studia Mathematica
PY - 2002
VL - 149
IS - 3
SP - 267
EP - 279
AB - The harmonic Cesàro operator is defined for a function f in $L^{p}(ℝ)$ for some 1 ≤ p < ∞ by setting $(f)(x): = ∫^{∞}_{x} (f(u)/u)du$ for x > 0 and $(f)(x): = -∫_{-∞}^{x} (f(u)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L¹_{loc}(ℝ)$ by setting $*(f)(x): = (1/x) ∫^{x₀} f(u)du$ 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 $f ∈ L^{p}(ℝ)$ for some 1 ≤ p ≤ 2, then $((f))^{∧}(t) = *(f̂)(t)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f ∈ L^{p}(ℝ)$ for some 1 < p ≤ 2, then $(*(f))^{∧}(t) = (f̂)(t)$ a.e. As a by-product of our proofs, we obtain representations of $((f))^{∧}(t)$ and $(*(f))^{∧}(t)$ in terms of Lebesgue integrals in case f belongs to $L^{p}(ℝ)$ for some 1 < p ≤ 2. These representations are valid for almost every t and may be useful in other contexts.
LA - eng
KW - harmonic Cesàro and Copson operators; Fourier transform; commuting relations; Hardy's inequalities; representations for Fourier transform
UR - http://eudml.org/doc/285082
ER -