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Weighted Bernstein-type inequalities, and embedding theorems for the model subspaces


Author: A. D. Baranov
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), nomer 5.
Journal: St. Petersburg Math. J. 15 (2004), 733-752
MSC (2000): Primary 47B32, 30B50
Published electronically: July 29, 2004
MathSciNet review: 2068792
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Abstract | References | Similar Articles | Additional Information

Abstract: Weighted estimates are obtained for the derivatives in the model (shift-coinvariant) subspaces $K^p_{\Theta}$, generated by meromorphic inner functions $\Theta$ of the Hardy class $H^p(\mathbb{C} ^+)$. It is shown that the differentiation operator acts from $K^p_{\Theta}$ to a space $L^p(w)$, where the weight $w$ depends on the function $\vert\Theta'\vert$, the rate of growth of the argument of $\Theta$ along the real line.

As an application of the weighted Bernstein-type inequalities, new Carleson-type theorems on embeddings of the subspaces $K^p_{\Theta}$ in $L^p(\mu)$ are proved. Also, results on the compactness of such embeddings are obtained, and properties of measures $\mu$ for which the norms $\Vert\cdot\Vert _{L^p(\mu)}$ and $\Vert\cdot\Vert _p$ are equivalent on a given model subspace $K^p_{\Theta}$, are established.


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  • 1. Paul Koosis, Introduction to 𝐻_{𝑝} spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
  • 2. N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223
  • 3. S. M. Nikol′skiĭ, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York-Heidelberg., 1975. Translated from the Russian by John M. Danskin, Jr.; Die Grundlehren der Mathematischen Wissenschaften, Band 205. MR 0374877
  • 4. K. M. D′yakonov, Entire functions of exponential type and model subspaces in 𝐻^{𝑝}, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 190 (1991), no. Issled. po Linein. Oper. i Teor. Funktsii. 19, 81–100, 186 (Russian, with English summary); English transl., J. Math. Sci. 71 (1994), no. 1, 2222–2233. MR 1111913, 10.1007/BF02111294
  • 5. Konstantin M. Dyakonov, Differentiation in star-invariant subspaces. I. Boundedness and compactness, J. Funct. Anal. 192 (2002), no. 2, 364–386. MR 1923406, 10.1006/jfan.2001.3920
  • 6. M. B. Levin, Estimation of the derivative of a meromorphic function on the boundary of the domain, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 24 (1975), 68–85, ii (Russian). MR 0402053
  • 7. Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
  • 8. Peter Borwein and Tamás Erdélyi, Sharp extensions of Bernstein’s inequality to rational spaces, Mathematika 43 (1996), no. 2, 413–423 (1997). MR 1433285, 10.1112/S0025579300011876
  • 9. Xin Li, R. N. Mohapatra, and R. S. Rodriguez, Bernstein-type inequalities for rational functions with prescribed poles, J. London Math. Soc. (2) 51 (1995), no. 3, 523–531. MR 1332889, 10.1112/jlms/51.3.523
  • 10. William S. Cohn, Carleson measures and operators on star-invariant subspaces, J. Operator Theory 15 (1986), no. 1, 181–202. MR 816238
  • 11. Konstantin M. Dyakonov, Smooth functions in the range of a Hankel operator, Indiana Univ. Math. J. 43 (1994), no. 3, 805–838. MR 1305948, 10.1512/iumj.1994.43.43035
  • 12. A. D. Baranov, The Bernstein inequality in the de Branges spaces and embedding theorems, Proceedings of the St. Petersburg Mathematical Society, Vol. IX, Amer. Math. Soc. Transl. Ser. 2, vol. 209, Amer. Math. Soc., Providence, RI, 2003, pp. 21–49. MR 2018371
  • 13. A. B. Aleksandrov, A simple proof of the Vol′berg-Treil′ theorem on the embedding of covariant subspaces of the shift operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 217 (1994), no. Issled. po Linein. Oper. i Teor. Funktsii. 22, 26–35, 218 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 85 (1997), no. 2, 1773–1778. MR 1327512, 10.1007/BF02355286
  • 14. Bill Cohn, Carleson measures for functions orthogonal to invariant subspaces, Pacific J. Math. 103 (1982), no. 2, 347–364. MR 705235
  • 15. A. L. Vol′berg and S. R. Treil′, Embedding theorems for invariant subspaces of the inverse shift operator, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (1986), no. Issled. Linein. Teor. Funktsii. XV, 38–51, 186–187 (Russian, with English summary); English transl., J. Soviet Math. 42 (1988), no. 2, 1562–1572. MR 849293, 10.1007/BF01665042
  • 16. A. B. Aleksandrov, Embedding theorems for coinvariant subspaces of the shift operator. II, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999), no. Issled. po Linein. Oper. i Teor. Funkts. 27, 5–48, 231 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 110 (2002), no. 5, 2907–2929. MR 1734326, 10.1023/A:1015379002290
  • 17. Alexei B. Aleksandrov, On embedding theorems for coinvariant subspaces of the shift operator. I, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., vol. 113, Birkhäuser, Basel, 2000, pp. 45–64. MR 1771751
  • 18. William S. Cohn, Radial imbedding theorems for invariant subspaces, Complex Variables Theory Appl. 17 (1991), no. 1-2, 33–42. MR 1123800
  • 19. A. L. Vol′berg, Thin and thick families of rational fractions, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 440–480. MR 643388
  • 20. V. N. Logvinenko and Ju. F. Sereda, Equivalent norms in spaces of entire functions of exponential type, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 20 (1974), 102–111, 175 (Russian). MR 0477719
  • 21. Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
  • 22. Douglas N. Clark, One dimensional perturbations of restricted shifts, J. Analyse Math. 25 (1972), 169–191. MR 0301534
  • 23. P. R. Ahern and D. N. Clark, Radial limits and invariant subspaces, Amer. J. Math. 92 (1970), 332–342. MR 0262511
  • 24. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • 25. A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 0177312
  • 26. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
  • 27. B. Ërikke and V. P. Khavin, Traces of harmonic functions and comparison of 𝐿^{𝑝}-norms of analytic functions, Math. Nachr. 123 (1985), 225–254 (Russian). MR 809347, 10.1002/mana.19851230120
  • 28. Joaquim Ortega-Cerdà and Kristian Seip, Fourier frames, Ann. of Math. (2) 155 (2002), no. 3, 789–806. MR 1923965, 10.2307/3062132
  • 29. M. B. Levin, An estimate of the derivative of a meromorphic function on the boundary of the domain, Dokl. Akad. Nauk SSSR 216 (1974), 495–497 (Russian). MR 0352468

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Additional Information

A. D. Baranov
Affiliation: St. Petersburg State University, Universitetskii Prospekt 28, Petrodvorets, St. Petersburg, 198504, Russia
Email: d.baranov@pop.ioffe.rssi.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00829-5
Keywords: Hardy class, inner function, shift-coinvariant subspace, Bernstein-type inequality
Received by editor(s): March 6, 2003
Published electronically: July 29, 2004
Additional Notes: The work was partially supported by RFBR grant no. 03-01-00377 and by the grant for Leading Scientific Schools no. NSH-2266.2003.1.
Article copyright: © Copyright 2004 American Mathematical Society