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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the weak type $ (1,\,1)$ inequality for conjugate functions


Author: Burgess Davis
Journal: Proc. Amer. Math. Soc. 44 (1974), 307-311
MSC: Primary 42A40
MathSciNet review: 0348381
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Abstract: A theorem of Kolmogorov states that there is a positive constant $ K$ such that if $ \tilde f$ is the conjugate function of an integrable real valued function $ f$ on the unit circle then $ m\{ \vert\tilde f\vert \geqq \lambda \} \leqq K\vert\vert f\vert{\vert _1}/\lambda ,\lambda > 0$. It is shown that the smallest possible value for $ K$ in this theorem, the so called weak type (1, 1) norm of the conjugate function operator, is $ (1 + {3^{ - 2}} + {5^{ - 2}} + \cdots )/(1 - {3^{ - 2}} - {5^{ - 2}} - \cdots ) \approx 1.347$. This number is also shown to be the weak type (1, 1) norm of the Hilbert transform operator on functions defined on the real line. The proof uses P. Levy's result that Brownian motion in the plane is conformally invariant.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0348381-6
PII: S 0002-9939(1974)0348381-6
Keywords: Conjugate function, weak-type inequality, Brownian motion
Article copyright: © Copyright 1974 American Mathematical Society