Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension

We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a positive answer for a class of functions that contains Sobolev functions on the real line which are decreasing away from the origin and even, and we provide a counterexample which is also decreasing away from the origin but not even.