A Modified Chang-Wilson-Wolff Inequality Via the Bellman Function
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We produce the optimal constant in an inequality bounding the exponential integral of a function by the exponential integral of its dyadic square function. This work is motivated by a well known result due to Chang, Wilson, and Wolff which controls the exponential integral of a function in terms of the essential supremum of its dyadic square function. Perhaps more interesting than the result itself is the method of proof. We establish our inequality and find the optimal constant using a Bellman function argument. This type of argument was pioneered in the 1980s by Donald Burkholder in his work on martingale transforms. Along the way, we trace the origin of the Bellman function technique back to Richard Bellman’s development of dynamic programming in the 1950s. Doing so provides some historical context and lends insight into the genesis of Burkholder’s ideas.