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A contractive Hardy-Littlewood inequality. (English) Zbl 1479.30043

The classical Hardy-Littlewood inequality [G. H. Hardy and J. E. Littlewood, Math. Ann. 97, 159–209 (1926; JFM 52.0267.01)] states that for \(f(z)=\sum_{n=0}^\infty a_nz^n\) in \(H^p(\mathbb T)\), \(0<p\le 2\), we have \[\sum_{n=0}^\infty \frac{|a_n|^2}{(n+1)^{2/p -1}} \le C_p \|f\|_p^2.\tag{1} \] In the paper the author proves the following.
Theorem 1.3. For each \(0<p\le 2\), there exists \(\epsilon_p>0\) such that for all \(f\in H^p(\mathbb T)\), \(f(z) = \sum_{n=0}^\infty a_n z^n\), we have \[ |a_0|^2 +\frac{p}2|a_1|^2 + \epsilon_p \sum_{n=0}^\infty \frac{|a_n|^2}{(n+1)^{2/p -1}} \le \|f\|^2_p. \] The inequality is sharp in the first two terms, but for \(n\ge 2\), the weight decays as in the Hardy-Littlewood inequality (1).

MSC:

30H10 Hardy spaces
30B10 Power series (including lacunary series) in one complex variable

Citations:

JFM 52.0267.01
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Full Text: DOI arXiv

References:

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