Multiplicative Inequalities for Functions from the Hardy Space $H^1$ and Their Application to the Estimation of Exponential Sums

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 $h\ge 3$.