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Logarithmic estimates for the Hilbert transform and the Riesz projection. (English) Zbl 1244.42013

Let \(\mathcal H\) be the Hilbert transform on the unit circle \(\mathbb T\) of the complex plane. As is well known by A. Zygmund, there are constants \(K\) and \(L\) such that \(\|{\mathcal H} f \|\leq K\|f \|_{L\log L} + L\) for all \(L\log L\)-functions \(f\) on \(\mathbb T\). For optimal values of the constants \(K\) and \(L\), a result of S. K. Pichorides [Stud. Math. 44, 165–179 (1972; Zbl 0238.42007)] states that to each \(K>2/\pi\) there corresponds a universal \(L=L(K)\) such that the aforementioned inequality holds for real-valued \(f\); on the other hand, such \(L\) does not exists when \(K\leq 2/\pi\).
In this paper the author extends the result of Pichorides to the \(\ell^2\)-valued \(L\log L\)-functions. Also, the author derives similar results for the Riesz projection and the co-analytic projection, using their relations to the Hilbert transform.

MSC:

42B30 \(H^p\)-spaces
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions

Citations:

Zbl 0238.42007
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References:

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