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• # Artykuł - szczegóły

## Studia Mathematica

2002 | 149 | 3 | 267-279

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

EN

### Abstrakty

EN
The harmonic Cesàro operator 𝓒 is defined for a function f in $L^{p}(ℝ)$ for some 1 ≤ p < ∞ by setting $𝓒(f)(x): = ∫^{∞}_{x} (f(u)/u)du$ for x > 0 and $𝓒(f)(x): = -∫_{-∞}^{x} (f(u)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L¹_{loc}(ℝ)$ by setting $𝓒*(f)(x): = (1/x) ∫^{x₀} f(u)du$ 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 $(𝓒(f))^{∧}(t) = 𝓒*(f̂)(t)$ a.e., where f̂ denotes the Fourier transform of f.
(ii) If $f ∈ L^{p}(ℝ)$ for some 1 < p ≤ 2, then $(𝓒*(f))^{∧}(t) = 𝓒(f̂)(t)$ a.e.
As a by-product of our proofs, we obtain representations of $(𝓒(f))^{∧}(t)$ and $(𝓒*(f))^{∧}(t)$ 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.

267-279

wydano
2002

### Twórcy

autor
• Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary