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On Cesàro means in Hardy spaces. (English) Zbl 0884.30028

Let \(\widehat{f}\) be the Fourier transformation of \(f\in L^1=L^1(0,2\pi)\) and let \(\sigma^\alpha_nf\) are the Cesàro means of order \(\alpha\) of \(f\).
In this paper author generalizes the Hardy-Littlewood theorem on \((C,\alpha)\) means in the Hardy spaces \(H^p\), \({1\over 2}<p<1\), \(\alpha>1/p-1\). More precisely, the main results of the paper are as follows:
Theorem. Let \(f\in L^1\) be such that \(\widehat{f}(k)=0\) for \(k < -2n\), where \(n\) is a positive integer.
(i) If \(1/2<p<1\) and \(\alpha>1/p-1\), then \(|\sigma^\alpha_nf|_p\leq c_{p,\alpha}|f|_p\).
(ii) If \(\alpha\geq 1\), then \(|\sigma^\alpha_nf|_{1/2}\leq c_\alpha(\log(2n))^2 |f|_{1/2}\).
(iii) If \(0<p<1/2\), then \(|\sigma^\alpha_nf|_p\leq c_{p,\alpha} n^{1/p-2} |f|_p\).
The estimates in (i) and (ii) are the best possible in the sense that there are (nontrivial) trigonometric polynomials \(f_n\), independent of \(p\), \(\alpha\), such that \(deg(f_n)\leq {n\over 2}\) and \(|\sigma^\alpha_nf_n|_{1/2}\geq c_\alpha |f_n|_{1/2} (\log n)^2\), for \(p={1\over 2}\), and \(|\sigma^\alpha_nf_n|_p\geq c_{p,\alpha} |f_n|_p n^{1/p-2}\),if \(0<p<{1\over 2}\).
Author also proved that the estimate (i) is true for all \(p<1\) if the Cesàro means are replaced by the Riesz (spherical) means.

MSC:

30D55 \(H^p\)-classes (MSC2000)
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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