(Sketch)

• •

  Bounded convex functions automatically have Lipschitz bound on slightly shrinked convex domains.

• •

  The second item follows from a slightly tricky application of mean value inequality for subharmonic functions, cf. [52, section 4.3].

• •

  The third item is because the function $\phi-\bar{\phi}$ has mean value zero, so an upper bound implies an $L^{1}$-bound, cf. [52, section 4.3].

• •

  One first apply the basic Skoda estimate Thm. 4.2 to the function $\phi$ on each logarithmic dyadic scale, where $\bar{\phi}$ is almost constant by the Lipschitz bound. Then we sum over all the logarithmic dyadic scales (cf. [52, section 4.6]).

∎