On the dimension of $p$-harmonic measure in space
John L. Lewis, Kaj Nyström, Andrew L. Vogel · Journal of the European Mathematical Society · 2013
Let \Omega\subset\mathbb {R}^{n} , n\geq 3 , and let p , 1 0 small such that if \Omega is a \delta -Reifenberg flat domain with \delta<\tilde\delta , then p -harmonic measure is concentrated on a set of \sigma -finite H^{n-1} -measure. We prove, for p \geq n , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p -harmonic measure is always less than n-1 . We also prove that if 2 , then there exist Wolff snowflakes such that the Hausdorff dimension of p -harmonic measure is less than n-1 , while if 1 , then there exist Wolff snowflakes such that the Hausdorff dimension of p -harmonic measure is larger than n-1 . Furthermore, perturbing off the case p = 2, we derive estimates when p is near 2 for the Hausdorff dimension of p -harmonic measure.