Abstract
Let ϕ be a unimodular function on the unit circle\(\mathbb{T}\) and let Kp(ϕ) denote the kernel of the Toeplitz operator Tϕ in the Hardy space Hp, p≥1;\(K_p (\varphi )\mathop = \limits^{def} \{ f \in H^p :T_\varphi f = 0\} \). Suppose Kp(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in Kp(ϕ). One of the main results is as follows.
Theorem 1
Let 1<p, q<+∞, 1<r≤+∞, q−1=p−1+r−1. Suppose |ϕ|≡1 on\(\mathbb{T}\) and ϕ∈W 1r (i.e.,\(\varphi ' \in L^r (\mathbb{T})\)). Then Kp(ϕ)⊂W 1q . Moreover, for any f∈Kp(ϕ) we have ‖f′‖q≤c(p, r)‖ϕ′‖r ‖f‖. Bibliography: 19 titles.
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Literature Cited
G. M. Airapetyan, “On the boundary values of functions that are generated by noncomplete systems of rational fractions with fixed poles,”Izv. Akad. Nauk Armenian SSR, Ser. Math.,23, No. 3, 297–301 (1988).
N. I. Akhiezer,Lectures on Approximation Theory, Nauka, Moscow (1965).
I. E. Verbitskii, “On Taylor coefficients andL p moduli of continuity of Blaschke products,”Zap. Nauchn. Semin. LOMI,107, 27–35 (1982).
J. B. Garnett,Bounded Analytic Functions, Academic Press, New York (1981).
K. Hoffman,Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, NJ (1962).
E. M. Dyn'kin, “Methods of the theory of singular integrals I,” in:Reviews of Science and Technics VINITI, Contemporary Problems of Mathematics [in Russian], Vol. 15, pp. 197–292.
K. M. D'yakonov, “Entire functions of exponential type and model subspaces inH p,”Zap. Nauchn. Semin. LOMI,190, 81–100 (1990).
K. M. Dyakonov, “Invariant subspaces of the backward shift operator in Hardy spaces,”Ph.D. Thesis, St. Petersburg State University, St. Petersburg (1991).
N. K. Nikol'skii,Treatise on the Shift Operator, Springer-Verlag, Berlin (1986).
S. M. Nikol'skii,Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, Berlin (1975).
A. A. Pekarskii, “Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation,”Mat. Sb.,124, No. 4, 571–588 (1984).
P. R. Ahern, “The mean modulus and the derivative of an inner function,”Indiana Univ. Math. J.,28, 311–348 (1979).
P. R. Ahern and D. N. Clark, “On inner functions withH p derivative,”Michigan Math. J.,21, 115–127 (1974).
A. P. Calderon, “Commutators of singular integral operators,”Proc. Natl. Acad. Sci. USA,53, 1092–1099 (1965).
W. S. Cohn, “Radial limits and star invariant subspaces of bounded mean oscillation,”Am. J. Math.,108, 719–749 (1986).
P. L. Duren, B. W. Romberg, and A. L. Shields, “Linear functionals onH p spaces with 0<p<1,”J. Reine Angew. Math.,238, 32–60 (1969).
E. Hayashi, “The kernel of a Toeplitz operator,” Integral Equations and Operator Theory,9, 588–591 (1986).
E. Hayashi, “Classification of nearly invariant subspaces of the backward shift,”Proc. Am. Math. Soc.,110, 441–448 (1990).
S. V. Hruscev and S. A. Vinogradov, “Inner functions and multipliers of Cauchy type integrals,”Ark. Mat.,19, 23–42 (1981).
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21.
Translated by K. M. D'yakonov.
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D'yakonov, K.M. Kernels of Toeplitz operators, smooth functions, and Bernstein type inequalities. J Math Sci 78, 131–141 (1996). https://doi.org/10.1007/BF02366031
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DOI: https://doi.org/10.1007/BF02366031