Given a function $f\in C([a,b])$, let $F(x) := \int_a^x f(y)\, dy$, $x\in [a,b]$, let $F^{**}$ be the convex envelope of $F$, and define $Tf := (F^{**})'$.
I would like to prove (or disprove) that $$ (I) \quad \int_a^b |Tf (y) - Tg(y)|\, dy \leq \int_a^b |f(y) - g(y)|\, dy, \qquad \forall f,g\in C([a,b]). $$ My tentative approach is based in trying to prove the following properties for the operator $T$:
(1) $\int_a^b Tf = \int_a^b f$ for every $f\in C([a,b])$;
(2) If $f\leq g$, then $Tf \leq Tg$.
Indeed, if (1) and (2) are true, then (I) will follow from a result by Crandall and Tartar.
Property (1) seems easy. I have problems in proving property (2), that seems correct on the base of some geometrical consideration.
Any advice (for proving (2) or for another approach towards (I)) will be highly appreciated.
EDIT: I have found a proof for (2), but it is long and tricky. I will write it only if there is someone interested.