Wolff snowflakes
Let \(D\) be a domain in \(\mathbb R^n\) (\(n\geq 3\)) and \(\mu\) be harmonic measure for \(D\) and some fixed point. Further, let \(\text{H-dim\,}\mu\) denote the infimum of those values \(k>0\) for which there exists \(E\subset \partial D\) with zero \(k\)-dimensional Hausdorff measure and \(\mu(E)=1\). \textit{T. H. Wolff} [Essays on Fourier analysis in honor of Elias M. Stein. Princeton Math. Ser. 42, 321--384 (1995; Zbl 0836.31004)] constructed certain domains \(D\) in \(\mathbb R^3\) with fractal-like boundaries, now called ``Wolff snowflakes'', for which either \(\text{H-dim\,}\mu >2\) or \(\text{H-dim}\;\mu <2 \). Let \(\widetilde{\mu }\) denote harmonic measure for \(\mathbb R^n\setminus\overline{D}\) and some fixed point. Then Wolff's construction actually leads to snowflakes in \(\mathbb R^n\) for which \(\text{H-dim }\mu >n-1\) and \(\text{H-dim}\;\widetilde{\mu}<n-1\); and these inequalities can be simultaneously reversed. The present paper verifies a conjecture of Wolff, that there are snowflakes for which \(\text{H-dim}\;\mu >n-1\) and \(\text{H-dim}\;\widetilde{\mu }>n-1\); and these inequalities can also be simultaneously reversed. The proofs are based on extensive difficult computations.
scientific article; zbMATH DE number 5117285