## On Kolmogorov’s inequalities $\tilde {f}_p \leq C_p$, $f_1$, $0<p<1$

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- by Burgess Davis PDF
- Trans. Amer. Math. Soc.
**222**(1976), 179-192 Request permission

## Abstract:

Let $\mu$ be a signed measure on the unit circle*A*of the complex plane satisfying $|\mu |(A) < \infty$, where $|\mu |(A)$ is the total variation of $\mu$, and let $\tilde \mu$ be the conjugate function of $\mu$. A theorem of Kolmogorov states that for each real number

*p*between 0 and 1 there is an absolute constant ${C_p}$ such that ${({(2\pi )^{ - 1}}\smallint _0^{2\pi }|\tilde \mu ({e^{i\theta }}){|^p}d\theta )^{1/p}} \leqslant {C_p}|\mu |(A)$. 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 $\nu$ is one of these measures the number ${({(2\pi )^{ - 1}}\smallint _0^{2\pi }|\tilde \nu ({e^{i\theta }}){|^p}d\theta )^{1/p}}/|\nu |(A)$ is the smallest possible value for ${C_p}$. These constants are also the best possible in the analogous Hilbert transform inequalities. The proof is based on probability theory.

## References

- D. L. Burkholder,
*Martingale transforms*, Ann. Math. Statist.**37**(1966), 1494–1504. MR**208647**, DOI 10.1214/aoms/1177699141 - Burgess Davis,
*On the weak type $(1,\,1)$ inequality for conjugate functions*, Proc. Amer. Math. Soc.**44**(1974), 307–311. MR**348381**, DOI 10.1090/S0002-9939-1974-0348381-6 - J. L. Doob,
*Semimartingales and subharmonic functions*, Trans. Amer. Math. Soc.**77**(1954), 86–121. MR**64347**, DOI 10.1090/S0002-9947-1954-0064347-X - Oliver Dimon Kellogg,
*Foundations of potential theory*, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR**0222317**, DOI 10.1007/978-3-642-86748-4 - H. P. McKean Jr.,
*Stochastic integrals*, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR**0247684** - 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). MR**312140**, DOI 10.4064/sm-44-2-165-179 - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**222**(1976), 179-192 - MSC: Primary 42A36; Secondary 60J65
- DOI: https://doi.org/10.1090/S0002-9947-1976-0422983-7
- MathSciNet review: 0422983