Multiplicative Inequalities for the~$L_1$ Norm: Applications in Analysis and Number Theory

S. V. Bochkarev

Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 55-70.

Résumé

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 $1\le \alpha \mathrm{\infty }$, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.

Bibliographie

[1] Zigmund A., Trigonometricheskie ryady, t. 1, 2, Mir, M., 1965 | MR

[2] Bochkarev S.V., “O koeffitsientakh Fure funktsii klassa $\mathrm{Lip}\alpha$ po polnym ortonormirovannym sistemam”, Mat. zametki, 7:4 (1970), 397–402 | Zbl

[3] Bochkarev S.V., “Absolyutnaya skhodimost ryadov Fure po polnym ortonormirovannym sistemam”, UMN, 27:2 (1972), 53–76 | MR | Zbl

[4] Zygmund A., “Remarque sur la convergence absolue des séries de Fourier”, J. London Math. Soc., 3 (1928), 194–196 | DOI | Zbl

[5] Salem R., “On a theorem of Zygmund”, Duke Math. J., 10 (1943), 23–31 | DOI | MR | Zbl

[6] Kahane J.-P., Séries de Fourier absolument convergentes, Ergebn. Math. und ihrer Grenzgebiete, 50, Springer, Berlin; New York, 1970 | MR | Zbl

[7] Wik I., “Criteria for absolute convergence of Fourier series of functions of bounded variation”, Trans. Amer. Math. Soc., 163 (1971), 1–24 | DOI | MR

[8] Bochkarev S.V., “O probleme Zigmunda”, Izv. AN SSSR. Cer. mat., 37:3 (1973), 630–638 | MR | Zbl

[9] Bochkarev S.V., “Ob absolyutnoi skhodimosti ryadov Fure po ogranichennym polnym ortonormirovannym sistemam funktsii”, Mat. sb., 93:2 (1974), 203–217 | Zbl

[10] Bochkarev S.V., Metod usrednenii v teorii ortogonalnykh ryadov i nekotorye voprosy teorii bazisov, Tr. MIAN, 146, Nauka, M., 1978 | MR | Zbl

[11] Bochkarev S.V., “Logarifmicheskii rost srednikh arifmeticheskikh ot funktsii Lebega ogranichennykh ortonormirovannykh sistem”, DAN SSSR, 223:1 (1975), 16–19 | MR | Zbl

[12] Olevskii A.M., Fourier series with respect to general orthogonal systems, Springer, Berlin, 1975 | MR | Zbl

[13] Bochkarev S.V., “Ob absolyutnoi skhodimosti ryadov Fure po ogranichennym sistemam”, Mat. zametki, 15:3 (1974), 363–370 | Zbl

[14] Bochkarev S.V., “Teorema Khausdorfa–Yunga–Rissa v prostranstvakh Lorentsa i multiplikativnye neravenstva”, Tr. MIAN, 219 (1997), 103–114 | MR | Zbl

[15] Konyagin S.V., “O probleme Littlvuda”, Izv. AN SSSR. Ser. mat., 45:2 (1981), 243–265 | MR | Zbl

[16] McGehee O.C., Pigno L., Smith B., “Hardy's inequality and the $L^1$ norm of exponential sums”, Ann. Math., 113:3 (1981), 613–618 | DOI | MR | Zbl

[17] Bochkarev S.V., “Ryady Valle Pussena v prostranstvakh BMO, $L_1$ i $H^1(D)$ i multiplikativnye neravenstva”, Tr. MIAN, 210, 1995, 41–64 | MR | Zbl

[18] Bochkarev S.V., “Novyi metod otsenki integralnoi normy eksponentsialnykh summ, prilozhenie k kvadratichnym summam”, DAN, 386:2 (2002), 156–159 | MR

[19] Bochkarev S.V., “Metod otsenki $L_1$-normy eksponentsialnoi summy na osnove arifmeticheskikh svoistv spektra”, Tr. MIAN, 232, 2001, 94–101 | MR | Zbl

[20] Bochkarev S.V., “Multiplikativnye neravenstva dlya funktsii iz prostranstva Khardi $H^1$ i ikh primenenie k otsenke eksponentsialnykh summ”, Tr. MIAN, 243, 2003, 96–103 | MR | Zbl

[21] Bochkarev S.V., “Novye multiplikativnye neravenstva i otsenki $L_1$-normy trigonometricheskikh ryadov i polinomov”, DAN, 404:6 (2005), 727–730 | MR | Zbl

[22] Kolmogoroff A.N., “Une série de Fourier–Lebesgue divergente presque partout”, Fund. math., 4 (1923), 324–328 | Zbl

[23] Bekkenbakh E., Bellman R., Neravenstva, Mir, M., 1965 | MR

[24] Hooley C., “On the representation of a number as the sum of two cubes”, Math. Ztschr., 82 (1963), 259–266 | DOI | MR | Zbl

[25] Hooley C., “On another sieve method and the numbers that are a sum of two $h$th powers”, Proc. London Math. Soc., 43:1 (1981), 73–109 | DOI | MR | Zbl

[26] Bochkarev S.V., “Novye neravenstva v teorii Littlvuda–Peli i otsenki $L_1$-normy trigonometricheskikh ryadov i polinomov”, Tr. MIAN, 248, 2005, 64–73 | MR | Zbl

[27] Konyagin S.V., “Ob otsenke $L_1$-normy odnoi eksponentsialnoi summy”, Teoriya priblizhenii funktsii i operatorov, Tez. dokl. Mezhdunar. konf. (Ekaterinburg, 2000), 88–89

[28] Garaev M.Z., “Upper bounds for the number of solutions of a Diophantine equation”, Trans. Amer. Math. Soc., 357:6 (2005), 2527–2534 | DOI | MR | Zbl

[29] Bochkarev S.V., “Multiplikativnye otsenki $L_1$-normy eksponentsialnykh summ”, Metricheskaya teoriya funktsii i smezhnye voprosy analiza, Izd-vo AFTs, M., 1999, 57–68 | MR