 Solution

Apply the derivative criterion for Hölder continuity up to a boundary to the estimate from part (c). For two points at distance $r$, move each vertically to height at least $r$, join them horizontally there, and move back. The two vertical integrals are bounded byand the horizontal integral is at most $Cr{r}^{-1+\alpha }=C{r}^{\alpha }$. Thus $h_T$ extends continuously to the bottom edge and satisfieson the half-rectangle. In particular, it is almost surely a Hölder continuous function there.