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The maximal conjugate Fejér operator is bounded from \(L^1\) to weak-\(L^1\). (English) Zbl 0927.47022

Summary: We consider the Fejér (or first arithmetic) means of the conjugate series to the Fourier series of a periodic function \(f\) integrable in Lebesgue’s sense on the torus \(\mathbb{T}:=[-\pi, \pi)\). A classical theorem of A. Zygmund says that the maximal conjugate Fejér operator \(\widetilde\sigma_*(f)\) is bounded from \(L^1(\mathbb{T})\) to \(L^p(\mathbb{T})\) for any \(0<p<1\). We sharpen this result by proving that \(\widetilde\sigma_*(f)\) is bounded from \(L^1(\mathbb{T})\) to weak-\(L^1(\mathbb{T})\). We prove an analogous result also for the Fejér means (or Riesz means of first-order) of the conjugate integral to the Fourier integral of a function \(f\) integrable in Lebesgue’s sense on the whole real line \(\mathbb{R}\).

MSC:

47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A50 Conjugate functions, conjugate series, singular integrals
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