Estimates of the singular numbers of the Carleson imbedding operator

O. G. Parfenov

Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 497-514

Résumé

Let $H^2$ be the Hardy class in the unit disc $D$ and $\mu$ a finite Borel measure in $D$. Carleson's theorem describes conditions on $\mu$ under which the corresponding imbedding operator $J\colon H^2\to L_2(\mu)$ (the Carleson operator) is bounded. From this theorem follows a criterion for compactness of $J$ in terms of $\mu$. This paper is devoted to further study of the Carleson operator. Almost sharp upper bounds on the singular numbers of $J$ are presented in terms of the intensity of $\mu$. For measures whose support is a set of nonzero linear measure adjacent to the unit circle (and when certain other conditions), an asymptotic formula is obtained. A study is begun of measures whose support has just one point on the unit circle. A solution of a problem from the theory of rational approximation, posed by A. A. Gonchar, is also presented. Bibliography: 17 titles.

Bibliographie

[1] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1970 | MR

[2] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[3] Fisher S. D., Charles Micchell, “The $n$-width of sets of analytic functions”, Duke Math. J., 47:4 (1980), 789–801 | DOI | MR | Zbl

[4] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra variatsionnykh zadach na resheniyakh ellipticheskikh sistem”, Zap. nauchn. sem. LOMI, 115 (1982), 23–39 | MR | Zbl

[5] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra variatsionnykh zadach na resheniyakh ellipticheskikh uravnenii”, Sib. matem. zhurn., 20:1 (1979), 3–22 | MR | Zbl

[6] Gonchar A. A., “Ratsionalnaya approksimatsiya”, Zap. nauchn. sem. LOMI, 81 (1978), 182–185

[7] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[8] Erokhin V. D., “O nailuchshei lineinoi approksimatsii funktsii, analiticheski prodolzhimykh s dannogo kontinuuma v dannuyu oblast”, UMN, 23 (1968), 93–135 | Zbl

[9] Akhiezer N. I., Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970 | MR | Zbl

[10] Parfenov O. G., “Poperechniki odnogo klassa analiticheskikh funktsii”, Matem. sb., 117(159) (1982), 279–285 | MR | Zbl

[11] Widom H., “Rational approximation and $n$-dimensional diameter”, J. Approxim. Theory, 5:2 (1972), 343–361 | DOI | MR | Zbl

[12] Uolsh Dzh. U., Interpolyatsiya i approksimatsiya ratsionalnymi funktsiyami v kompleksnoi oblasti, IL, M., 1961 | MR

[13] Adamyan V. M., Arov D. Z., Krein M. G., “Analiticheskie svoistva par Shmidta gankeleva operatora i obobschennaya zadacha Shura–Takagi”, Matem. sb., 86(128):1(9) (1971), 34–75 | MR | Zbl

[14] Peller V. V., “Operatory Gankelya klassa $\gamma_p$ i ikh prilozheniya (ratsionalnaya approksimatsiya, gaussovskie protsessy, problemy mazhoratsii operatorov)”, Matem. sb., 113(155) (1980), 538–581 | MR | Zbl

[15] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[16] Parfenov O. G., “Asimptotika singulyarnykh chisel operatorov vlozheniya nekotorykh klassov analiticheskikh funktsii”, Matem. sb., 115(157) (1981), 632–641 | MR | Zbl

[17] Zakharyuta V. P., Skiba N. I., “Otsenki $n$-poperechnikov nekotorykh klassov funktsii, analiticheskikh na rimanovykh poverkhnostyakh”, Matem. zametki, 19:6 (1976), 899–911 | MR | Zbl