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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A weak-type inequality for
differentially subordinate harmonic functions


Author: Changsun Choi
Journal: Trans. Amer. Math. Soc. 350 (1998), 2687-2696
MSC (1991): Primary 31B05, 31B15; Secondary 42A50, 60G42
DOI: https://doi.org/10.1090/S0002-9947-98-02259-4
MathSciNet review: 1617340
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Abstract: Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder $\mu(|v|\ge 1)\le 2\|u\|_1$ for two harmonic functions $u$ and $v$. That is, we prove the sharp weak-type inequality $\mu(|v|\ge 1)\le K\|u\|_1$ under the assumptions that $|v(\xi)|\le |u(\xi)|$, $|\nabla v|\le|\nabla u|$ and the extra assumption that $\nabla u\cdot\nabla v=0$. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant $K$ is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.


References [Enhancements On Off] (What's this?)

  • 1. S. Axler et al., Harmonic function theory, Springer-Verlag, 1992. MR 93f:31001
  • 2. A. Baernstein II, Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833-852. MR 80g:30022
  • 3. R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J. 80 (1995), 575-600. MR 96k:60108
  • 4. D. L. Burkholder, Differential subordination of harmonic functions and martingales, Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), Lecture Notes in Mathematics 1384 (1989), 1-23. MR 90k:31004
  • 5. D. L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), 995-1025. MR 95k:60085
  • 6. B. Davis, On the weak type $(1,1)$ inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307-311. MR 50:879
  • 7. T. W. Gamelin, Uniform algebras and Jensen measures, Cambridge University Press, London, 1978.
  • 8. Y. Katznelson, An introduction to harmonic analysis, Dover Publications, 1976. MR 54:10976
  • 9. A. N. Kolmogorov, Sur les fonctions harmoniques conjugées et les séries de Fourier, Fund. Math. 7 (1925), 24-29.
  • 10. 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. MR 47:702
  • 11. M. Riesz, Sur les fonctions conjugées, Math. Z. 27 (1927), 218-244.

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Additional Information

Changsun Choi
Affiliation: Department of Mathematics, University of Illinois, 273 Altgeld Hall, 1409 West Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, KAIST Taejon, 305-701 Korea
Email: cschoi@math.kaist.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-98-02259-4
Keywords: Harmonic functions, harmonic measure, differential subordination, weak-type inequality, Burkholder's inequality, Kolmogorov's inequality, Davis's constant
Received by editor(s): October 7, 1995
Article copyright: © Copyright 1998 American Mathematical Society

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