  We establish a new oscillation estimate for solutions of nonlinear partial
differential equations of elliptic, degenerate type. This new tool yields a
precise control on the growth rate of solutions near their set of critical
points, where ellipticity degenerates. As a consequence, we are able to prove
the planar counterpart of the longstanding conjecture that solutions of the
degenerate $p$-Poisson equation with a bounded source are locally of class
$C^{p^\prime}=C^{1,\frac{1}{p-1}}$; this regularity is optimal.
