Loukas Grafakos, Stephen Montgomery-Smith and Olexei Motrunich, A sharp estimate for the Hardy-Littlewood maximal function.
Studia Math, 134, (1999), 57-67.
The best constant in the usual \(L^p\) norm inequality for the centered Hardy-Littlewood maximal function on \(\mathbb R^1\) is obtained for the class of all "peak-shaped" functions. A positive function on the line is called "peak-shaped" if it is positive and convex except at one point. The techniques we use include convexity and an adaptation of the standard Euler-Langrange variational method.
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