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 Probl. Anal. Issues Anal., 2013, Volume 2(20), Issue 2, Pages 21–58 (Mi pa6)

On a generalization of an inequality of Bohr

B. F. Ivanov

St. Petersburg State Technological University for Plant and Polymers

Abstract: Let $p\in (1, 2], n\ge 1, S\subseteq R^{n}$ and $\Gamma(S, p)$— the set of all functions, $\gamma(t)\in L ^{p}(R ^{n})$ the support of the Fourier transform of which lies in $S$. We obtain the inequality conditions $||\int \limits_{E_t}\gamma(\tau)d\tau|| _{L ^{\infty}(R^n)}\le C||\gamma(\tau)|| _{L ^{p}(R^n)}$, where $t=(t _{1}, t _{2}, …, t _{n})\in R^{n}, E _{t} = \{\tau|\tau=(\tau _{1},\tau _{2},…,\tau _{n})\in R^{n}, \tau_j\in [0,t_j]$, if $t_j\ge 0$ and $\tau_{j}\in (t_j,0]$, if $\tau_{j}< 0, 1\le j\le n\}, \gamma(\tau)\in \Gamma(S,p)$ and constant $C$ does not depend on $\gamma(t)$. Also were considered some validity conditions on the inequality on non-trivial subsets $\Gamma(S, p)$ in cases, where they were not satisfied on the whole $\Gamma(S, p)$.

Keywords: inequality of Bohr

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Bibliographic databases:
UDC: 517
MSC: 26D99

Citation: B. F. Ivanov, “On a generalization of an inequality of Bohr”, Probl. Anal. Issues Anal., 2(20):2 (2013), 21–58

Citation in format AMSBIB
\Bibitem{Iva13} \by B.~F.~Ivanov \paper On a generalization of an inequality of Bohr \jour Probl. Anal. Issues Anal. \yr 2013 \vol 2(20) \issue 2 \pages 21--58 \mathnet{http://mi.mathnet.ru/pa6} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3168876} \zmath{https://zbmath.org/?q=an:1292.26050}