Estimates and structure of \(\alpha\)-harmonic functions
The authors first prove a uniform boundary Harnack principle for nonnegative harmonic function for the fractional Laplacian in an arbitrary domain \(D\subset \mathbb{R}^d\), thus extending the result of \textit{R.~Song} and \textit{J.-M.~Wu} [J.~Funct.~Anal.~168, No. 2, 403--427 (1999; Zbl 0945.31006)], where the boundary Harnack principle was proved with the constant depending on the domain. Independence of the constant on the domain yields a unique representation of nonnegative harmonic functions as integrals against measures on \(D^c\cup \{\infty\}\) satisfying an integrability condition. The authors also identify the Martin boundary of \(D\) as the part of the Euclidean boundary consisting of the so called accessible points.
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