Inner functions with derivatives in the weak Hardy space

Cima, Joseph; Nicolau, Artur

doi:10.1090/S0002-9939-2014-12305-7

Inner functions with derivatives in the weak Hardy space
HTML articles powered by AMS MathViewer

by Joseph A. Cima and Artur Nicolau

Proc. Amer. Math. Soc. 143 (2015), 581-594

DOI: https://doi.org/10.1090/S0002-9939-2014-12305-7

Published electronically: October 1, 2014

PDF | Request permission

Abstract:

It is proved that exponential Blaschke products are the inner functions whose derivative is in the weak Hardy space. As a consequence, it is shown that exponential Blaschke products are Frostman shift invariant. Exponential Blaschke products are described in terms of their logarithmic means and also in terms of the behavior of the derivatives of functions in the corresponding model space.

References

Patrick Ahern, The mean modulus and the derivative of an inner function, Indiana Univ. Math. J. 28 (1979), no. 2, 311–347. MR 523107, DOI 10.1512/iumj.1979.28.28022
P. R. Ahern and D. N. Clark, On inner functions with $H^{p}$-derivative, Michigan Math. J. 21 (1974), 115–127. MR 344479
A. B. Aleksandrov, $A$-integrability of boundary values of harmonic functions, Mat. Zametki 30 (1981), no. 1, 59–72, 154 (Russian). MR 627941
A. B. Aleksandrov, Essays on nonlocally convex Hardy classes, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 1–89. MR 643380
Christopher J. Bishop, An indestructible Blaschke product in the little Bloch space, Publ. Mat. 37 (1993), no. 1, 95–109. MR 1240926, DOI 10.5565/PUBLMAT_{3}7193_{0}8
James G. Caughran and Allen L. Shields, Singular inner factors of analytic functions, Michigan Math. J. 16 (1969), 409–410. MR 252648
Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. MR 2215991, DOI 10.1090/surv/125
William S. Cohn, On the $H^{p}$ classes of derivatives of functions orthogonal to invariant subspaces, Michigan Math. J. 30 (1983), no. 2, 221–229. MR 718268
Michael R. Cullen, Derivatives of singular inner functions, Michigan Math. J. 18 (1971), 283–287. MR 283205
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Konstantin M. Dyakonov, Smooth functions in the range of a Hankel operator, Indiana Univ. Math. J. 43 (1994), no. 3, 805–838. MR 1305948, DOI 10.1512/iumj.1994.43.43035
O. Frostman, Potential d’equilibre et capacité des ensembles avec quelques applications á la théorie des fonctions, Medd. Lunds Univ. Mat. Sem. 3 (1935), 1–118.
Emmanuel Fricain and Javad Mashreghi, Integral means of the derivatives of Blaschke products, Glasg. Math. J. 50 (2008), no. 2, 233–249. MR 2417618, DOI 10.1017/S0017089508004175
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Daniel Girela, Cristóbal González, and Miroljub Jevtić, Inner functions in Lipschitz, Besov, and Sobolev spaces, Abstr. Appl. Anal. , posted on (2011), Art. ID 626254, 26. MR 2802834, DOI 10.1155/2011/626254
Daniel Girela, José Ángel Peláez, and Dragan Vukotić, Integrability of the derivative of a Blaschke product, Proc. Edinb. Math. Soc. (2) 50 (2007), no. 3, 673–687. MR 2360523, DOI 10.1017/S0013091504001014
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
Javad Mashreghi, Derivatives of inner functions, Fields Institute Monographs, vol. 31, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013. MR 2986324, DOI 10.1007/978-1-4614-5611-7
Renate McLaughlin, Exceptional sets for inner functions, J. London Math. Soc. (2) 4 (1972), 696–700. MR 296309, DOI 10.1112/jlms/s2-4.4.696
H. Stephen Morse, Destructible and indestructible Blaschke products, Trans. Amer. Math. Soc. 257 (1980), no. 1, 247–253. MR 549165, DOI 10.1090/S0002-9947-1980-0549165-3
José Ángel Peláez, Sharp results on the integrability of the derivative of an interpolating Blaschke product, Forum Math. 20 (2008), no. 6, 1039–1054. MR 2479288, DOI 10.1515/FORUM.2008.046
David Protas, Blaschke products with derivative in $H^{p}$ and $B^{p}$, Michigan Math. J. 20 (1973), 393–396. MR 344478
David Protas, Mean growth of the derivative of a Blaschke product, Kodai Math. J. 27 (2004), no. 3, 354–359. MR 2100928, DOI 10.2996/kmj/1104247356
David Protas, Blaschke products with derivative in function spaces, Kodai Math. J. 34 (2011), no. 1, 124–131. MR 2786785, DOI 10.2996/kmj/1301576766
William T. Ross, Indestructible Blaschke products, Banach spaces of analytic functions, Contemp. Math., vol. 454, Amer. Math. Soc., Providence, RI, 2008, pp. 119–134. MR 2408240, DOI 10.1090/conm/454/08832
I. È. Verbitskiĭ, On Taylor coefficients and $L_{p}$-moduli of continuity of Blaschke products, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 107 (1982), 27–35, 228 (Russian, with English summary). Investigations on linear operators and the theory of functions, X. MR 676147
I. È. Verbitskiĭ, Inner functions, Besov spaces and multipliers, Dokl. Akad. Nauk SSSR 276 (1984), no. 1, 11–14 (Russian). MR 744877