Geodesic
Multiplicative Inequalities for the~$L_1$ Norm: Applications in Analysis and Number Theory
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 55-70

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The paper is devoted to multiplicative lower estimates for the $L_1$ norm and their applications in analysis and number theory. Multiplicative inequalities of the following three types are considered: martingale (for the Haar system), complex trigonometric (for exponential sums), and real trigonometric. A new method for obtaining sharp bounds for the integral norm of trigonometric and power series is proposed; this method uses the number-theoretic and combinatorial characteristics of the spectrum. Applications of the method (both in $H^1$ and $L_1$) to an important class of power density spectra, including $[n^\alpha]$ with $1\le\alpha \infty$, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions. 
@article{TM_2006_255_a4,
     author = {S. V. Bochkarev},
     title = {Multiplicative {Inequalities} for the~$L_1$ {Norm:} {Applications} in {Analysis} and {Number} {Theory}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {55--70},
     publisher = {mathdoc},
     volume = {255},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_255_a4/}
}
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S. V. Bochkarev. Multiplicative Inequalities for the~$L_1$ Norm: Applications in Analysis and Number Theory. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 55-70. http://geodesic.mathdoc.fr/item/TM_2006_255_a4/