On a theorem of Privalov and normal functions

Author:
Daniel Girela

Journal:
Proc. Amer. Math. Soc. **125** (1997), 433-442

MSC (1991):
Primary 30D45, 30D55

DOI:
https://doi.org/10.1090/S0002-9939-97-03544-2

MathSciNet review:
1363422

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

**[1]**J. M. Anderson, J. Clunie and Ch. Pommerenke,*On Bloch functions and normal functions*, J. Reine Angew. Math.**270**(1974), 12-37. MR**50:13536****[2]**A. Baernstein and J. E. Brown,*Integral means of derivatives of monotone slit mappings*, Comment. Math. Helvetici**57**(1982), 331-348. MR**85j:30033****[3]**C. Bennett and M. Stoll,*Derivatives of analytic functions and bounded mean oscillation*, Arch. Math.**47**(1986), 438-442. MR**88a:30074****[4]**R. D. Berman, L. Brown and W. S. Cohn,*Moduli of continuity and generalized BCH sets*, Rocky Mountain J. Math.**17**(3) (1987), 315-338. MR**88j:30010****[5]**O. Blasco and G. Soares de Souza,*Spaces of analytic functions on the disc where the growth of depends on a weight*, J. Math. Anal. Appl.**147**(2) (1990), 580-598. MR**92e:46051****[6]**P. Bourdon, J. Shapiro and W. Sledd,*Fourier series, mean Lipschitz spaces and bounded mean oscillation*, Analysis at Urbana 1, Proceedings of the Special Year in Modern Analysis at the University of Illinois, 1986-87 (E. R. Berkson, N. T. Peck and J. Uhl, eds.), London Math. Soc. Lecture note series 137, Cambridge Univ. Press, 1989, pp. 81-110. MR**90j:42011****[7]**J. A. Cima and K. E. Petersen,*Some analytic functions whose boundary values have bounded mean oscillation*, Math. Z.**147**(1976), 237-247. MR**53:8431****[8]**P. L. Duren,*Theory of spaces*, Academic Press (New York), 1970. MR**42:3552****[9]**O. Lehto and K. J. Virtanen,*Boundary behaviour and normal meromorphic functions*, Acta Math.**97**(1957), 47-65. MR**19:403****[10]**J. Lippus,*On multipliers preserving the classes of functions with a given major of the modulus of continuity*, J. Approx. Th.**66**(1991), 190-197. MR**92m:42010****[11]**K. I. Oskolkov,*An estimate of the rate of approximation of a continuous function and its conjugate by Fourier sums on a set of total measure*, Izv. Acad. Nauk SSSR Ser. Mat.**38**(1974), no.6, 1393-1407 (Russian); English transl. in Math. USSR Izv.**8**(6) (1974), no. 6, 1372-1386 (1976). MR**50:10663****[12]**K. I. Oskolkov,*Uniform modulus of continuity of summable functions on sets of positive measure*, Dokl. Akad. Nauk SSSR**229**(1976), no.2, 304-306 (Russian); English transl. in Sov. Math. Dokl.**17**(1976), no. 4, 1028-1030 (1977). MR**57:9917****[13]**K. I. Oskolkov,*Approximation properties of summable functions on sets of full measure*, Mat. Sbornik, n. Ser.**103(145)**(1977), 563-589 (Russian); English transl. in Math. USSR Sbornik**32**(1977), no. 4, 489-514 (1978). MR**57:13343****[14]**K. I. Oskolkov,*On Luzin's C-property for a conjugate function*, Trudy Mat. Inst. Steklova**164**(1983), 124-135 (Russian); English transl. in Proc. Steklov Inst. Math.**164**(1985), 141-153. MR**86e:42019****[15]**Ch. Pommerenke,*Univalent Functions*, Vandenhoeck und Ruprecht, Göttingen, 1975. MR**58:22526****[16]**D. Protas,*Blaschke products with derivative in and*, Michigan Math. J.**20**(1973), 393-396. MR**49:9217****[17]**S. Yamashita,*A non-normal function whose derivative has finite area integral of order*, Ann. Acad. Sci. Fenn. Ser. A I Math.**4**(2) (1979), 293-398. MR**81k:34002****[18]**S. Yamashita,*A non-normal function whose derivative is of Hardy class ,*, Canad. Math. Bull.**23**(4) (1980), 499-500. MR**82a:34006**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
30D45,
30D55

Retrieve articles in all journals with MSC (1991): 30D45, 30D55

Additional Information

**Daniel Girela**

Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

Email:
Girela@ccuma.sci.uma.es

DOI:
https://doi.org/10.1090/S0002-9939-97-03544-2

Keywords:
Normal functions,
Hardy spaces,
integral means,
theorem of Privalov

Received by editor(s):
November 1, 1994

Received by editor(s) in revised form:
June 25, 1995

Additional Notes:
This research has been supported in part by a D.G.I.C.Y.T. grant (PB91-0413) and by a grant from “La Junta de Andalucía”

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1997
American Mathematical Society