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

Ferenc Móricz

Studia Mathematica, Tome 151 (2002), p. 267-279 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2002-01-01

Zbl 1011.47026

EUDML-ID : urn:eudml:doc:285082