On Kolmogorov's inequalities , ,

Author:
Burgess Davis

Journal:
Trans. Amer. Math. Soc. **222** (1976), 179-192

MSC:
Primary 42A36; Secondary 60J65

MathSciNet review:
0422983

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a signed measure on the unit circle *A* of the complex plane satisfying , where is the total variation of , and let be the conjugate function of . A theorem of Kolmogorov states that for each real number *p* between 0 and 1 there is an absolute constant such that . Here it is shown that measures putting equal and opposite mass at points directly opposite from each other on the unit circle, and no mass any place else, are extremal for all of these inequalities, that is, if is one of these measures the number is the smallest possible value for . These constants are also the best possible in the analogous Hilbert transform inequalities. The proof is based on probability theory.

**[1]**D. L. Burkholder,*Martingale transforms*, Ann. Math. Statist.**37**(1966), 1494–1504. MR**0208647****[2]**Burgess Davis,*On the weak type (1,1) inequality for conjugate functions*, Proc. Amer. Math. Soc.**44**(1974), 307–311. MR**0348381**, 10.1090/S0002-9939-1974-0348381-6**[3]**J. L. Doob,*Semimartingales and subharmonic functions*, Trans. Amer. Math. Soc.**77**(1954), 86–121. MR**0064347**, 10.1090/S0002-9947-1954-0064347-X**[4]**Oliver Dimon Kellogg,*Foundations of potential theory*, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. MR**0222317****[5]**H. P. McKean Jr.,*Stochastic integrals*, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR**0247684****[6]**S. K. Pichorides,*On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov*, Studia Math.**44**(1972), 165–179. (errata insert). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, II. MR**0312140****[7]**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
42A36,
60J65

Retrieve articles in all journals with MSC: 42A36, 60J65

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0422983-7

Keywords:
Conjugate function,
Brownian motion,
Kolmogorov's inequalities

Article copyright:
© Copyright 1976
American Mathematical Society