The s-function and the exponential integral
by
Leonid Slavin
University of Missouri-Columbia

The order of exponential integrability is an important characteristic of many function spaces. Two prominent examples are the John-Nirenberg and Chang-Wilson-Wolff inequalities

(JN)    〈ej-〈j〉I〉I ≤ A(∥j∥BMO),        (CWW)    〈eb(j-〈j〉I)2〉I ≤ B(∥Sj∥L∞(I)),

where 〈j〉_I is the average of a function j over an interval I and Sj is the square function of j relative to I. The inequalities are valid for certain ranges of ∥j∥_BMO, ∥Sj∥_L^∞(I), and b. In recent years, the Bellman function method has efficiently yielded sharp results in both inequalities (including explicit expressions for A and B). In this talk, we prove the new sharp estimate

〈ej-〈j〉I〉I ≤ 〈e[1/2](Sj)2〉I

and use it to deduce the previously known, as well as new cases of exponential integrability.

PDF

Date received: February 16, 2008