Skip to main content
SpringerLink
Log in

On a maximum principle for pseudocontinuable functions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Let Θ be an inner function and let α∈ℂ, |α|=1 Denote by σα the nonnegative singular measure whose Poisson integral is equal to\(\operatorname{Re} \frac{{\alpha + \Theta }}{{\alpha - \Theta }}\). A theorem of Clark provides a natural unitary operator Uα that identifies H2 ⊝Θ H2 with L2(σα). The following fact is established. Assume that f∈H2⊝Θ H2, 2<p≤+∞, α≠β. Then

$$\left\| f \right\|_{H^p } \leqslant C\left( {\alpha ,\beta ,p} \right)\left( {\left\| {U_\alpha f} \right\|_{L^p (\sigma _a )} + \left\| {U_B f} \right\|_{L^p (\sigma _\beta )} } \right)$$

.

Bibliography:11 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature Cited

  1. J. B. Garnett,Bounded Analytic Functions, Academic Press (1981).

  2. N. K. Nikol'skii,Lectures on the Shift Operator [in Russian], Moscow (1980).

  3. W. Cohn, “Carleson measures for functions orthogonal to invariant subspaces,”Pac. J. Math.,103, 367–384 (1982).

    Google Scholar 

  4. W. Cohn, “Radial limits and star invariant subspaces of bounded mean oscillation,”Amer. J. Math.,108, 719–749 (1986).

    MATH  MathSciNet  Google Scholar 

  5. A. L. Vol'berg and S. R. Treil', “Embedding theorems for invariant subspaces of the inverse shift operator,”Zap. Nauchn. Semin. LOMI,149, 38–51 (1986).

    Google Scholar 

  6. W. Cohn, “A maximal principle for star invariant subspaces,”Houston J. Math.,14, 23–37 (1988).

    MATH  MathSciNet  Google Scholar 

  7. A. B. Aleksandrov, “Inner functions and the related spaces of pseudocontinuable functions,”Zap. Nauch. Semin. LOMI,170, 7–33 (1989).

    MATH  Google Scholar 

  8. K. M. D'yakonov, “Smooth functions and coinvariant subspaces of the shift operator,”Algebra Analiz,4, 117–147 (1992).

    MATH  MathSciNet  Google Scholar 

  9. A. G. Poltoratskii, “Boundary behavior of pseudocontinuable functions,”Algebra Analiz,5, 189–210 (1993).

    MATH  MathSciNet  Google Scholar 

  10. D. Clark, “One dimensional perturbations of restricted shifts,”J. Anal. Math.,25, 169–191 (1972).

    MATH  Google Scholar 

  11. S. V. Khrushchev and S. A. Vinogradov, “Inner functions and multipliers of Cauchy type integrals,”Ark. Mat.,19, 23–42 (1981).

    MathSciNet  Google Scholar 

Download references

Authors
  1. A. B. Aleksandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromZapiski Nauchnykh Seminarou POMI, Vol. 217, 1994, pp. 16–25

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aleksandrov, A.B. On a maximum principle for pseudocontinuable functions. J Math Sci 85, 1767–1772 (1997). https://doi.org/10.1007/BF02355285

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02355285

Keywords

Search

Navigation