The Bellman functions of dyadic-like maximal operators and related inequalities

By A. Melas. 2005

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Monge--Amp\`ere equation and Bellman optimization of Carleson Embedding Theorems (2008, methodology): “CET is deﬁned in ΩCET := {x = (x 1,x 2,x 3): x 2 ≤ x 2, m ≤ x 3 ≤ M}. (1.4) We introduce also B(x 1,x 2,x 3). This is what is often called the Bellman function for the Carleson Embedding Theorem, see [M], [NT]. B(x 1,x 2,x 3) := sup 1 |I| X J∈D(I) |hφi J |2α J : hφi I = x 1, hφ2i I = x 2, 1 |I| X J∈”
Sharp results in the integral-form John-Nirenberg inequality (2007, background): “Besides [13], we note the papers [5, 15, 11]; several others are in the works.”
The foundational inequalities of D.L. Burkholder and some of their ramifications (2010, methodology): “For more on this, see [110], [111], [152], and specially the recent papers [151], [153] which contain a treatment, based on the Monge-Ampère equation, on how to solve many Bellman equations, including Burkholder’s.”