An example on null sets of parabolic measures

Abstract:

A set $E$ on the real line of Hausdorff dimension 1 is constructed, such that the graph of $E$ on the boundary of any ${C^2}$ domain $\left \{ {t > \tau (x)} \right \}$, or on the boundary of any ${\text {Lip}}\tfrac {1}{2}$ domain $\left \{ {x > \chi (t)} \right \}$, is null with respect to the parabolic measure associated with any parabolic operator $L = a(x,t){\partial ^2}/\partial {x^2} - \partial /\partial t$ on ${{\mathbf {R}}^2};a$ is Hölder continuous and $0 < {\Lambda _1} \leq a \leq {\Lambda _2} < \infty$ for some constants ${\Lambda _1}$ and ${\Lambda _2}$.