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

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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 $h\ge 3$. 
@article{TM_2003_243_a8,
     author = {S. V. Bochkarev},
     title = {Multiplicative {Inequalities} for {Functions} from the {Hardy} {Space~}$H^1$ and {Their} {Application} to the {Estimation} of {Exponential} {Sums}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {96--103},
     publisher = {mathdoc},
     volume = {243},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_243_a8/}
}
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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/