We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure | E | of such a set. We exploit a rational change of variable of the type used by George Boole in his paper "On the comparison of transcendents, with certain applications to the theory of definite integrals" together with the observation that if two functions f and g have the same Lp norm in a range of exponents p1 < p < p2 then their distribution functions coincide. © 2009 Elsevier Inc. All rights reserved.
Colzani, L., Laeng, E., Monzon, L. (2010). Variations on a theme of Boole and Stein-Weiss. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 363(1), 225-229 [10.1016/j.jmaa.2009.08.017].
Variations on a theme of Boole and Stein-Weiss
COLZANI, LEONARDO;
2010
Abstract
We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure | E | of such a set. We exploit a rational change of variable of the type used by George Boole in his paper "On the comparison of transcendents, with certain applications to the theory of definite integrals" together with the observation that if two functions f and g have the same Lp norm in a range of exponents p1 < p < p2 then their distribution functions coincide. © 2009 Elsevier Inc. All rights reserved.
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