Title: Sharp total variation results for maximal functions,
Abstract: In this article, we prove some total variation inequalities for maximal
functions. Our results deal with two possible generalizations of the results
contained in Aldaz and Pérez Lázaro's work, one of whose considers a
variable truncation of the maximal function, and the other one interpolates the
centered and the uncentered maximal functions. In both contexts, we find sharp
constants for the desired inequalities, which can be viewed as progress towards
the conjecture that the best constant for the variation inequality in the
centered context is one. We also provide counterexamples showing that our
methods do not apply outside the stated parameter ranges.