Boundedness of a variation of the positive harmonic function along the normals to the boundary

Abstract:

Let $u$ be a positive harmonic function on the unit disk. Bourgain showed that the radial variation \[ \mathrm {var}(u\vert _{[0,re^{i\theta }]}) = \int _0^1|u’(re^{i\theta })| dr \] of $u$ is finite for many points $\theta$, and moreover, that the set \[ \mathcal {V}(u)=\big \{e^{i\theta } : \mathrm {var}\big (u\vert _{[0,re^{i\theta }]}\big )< +\infty \big \} \] is dense in the unit circle $\mathbb {T}$ and its Hausdorff dimension equals one. In the paper, this result is generalized to a class of smooth domains in $\mathbb {R}^d$, $d\geq 3$.