Geodesic
Multiplicative Inequalities for Functions from the Hardy Space $H^1$ and Their Application to the Estimation of Exponential Sums
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 96-103 
S. V. Bochkarev. Multiplicative Inequalities for Functions from the Hardy Space $H^1$ and Their Application to the Estimation of Exponential Sums. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 96-103. http://geodesic.mathdoc.fr/item/TM_2003_243_a8/
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     author = {S. V. Bochkarev},
     title = {Multiplicative {Inequalities} for {Functions} from the {Hardy} {Space~}$H^1$ and {Their} {Application} to the {Estimation} of {Exponential} {Sums}},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Multiplicative inequalities are established for functions from the Hardy space $H^1$; based on these inequalities, lower estimates are found for the $L_1$-norm of a general exponential sum. Estimates for the $L_1$-norm of quadratic sums and sums with a power-law spectrum $\{n^h\}$, $h\ge 3$, are derived under certain conditions imposed on the absolute values of the coefficients in the sums. The estimates are sharp for $h\ge 3$.

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