On the weak type inequality for conjugate functions

Author:
Burgess Davis

Journal:
Proc. Amer. Math. Soc. **44** (1974), 307-311

MSC:
Primary 42A40

DOI:
https://doi.org/10.1090/S0002-9939-1974-0348381-6

MathSciNet review:
0348381

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Abstract: A theorem of Kolmogorov states that there is a positive constant such that if is the conjugate function of an integrable real valued function on the unit circle then . It is shown that the smallest possible value for in this theorem, the so called weak type (1, 1) norm of the conjugate function operator, is . 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:
https://doi.org/10.1090/S0002-9939-1974-0348381-6

Keywords:
Conjugate function,
weak-type inequality,
Brownian motion

Article copyright:
© Copyright 1974
American Mathematical Society