Harnack inequality for symmetric stable processes on fractals (Q1608681)

The authors consider nonnegative harmonic functions of symmetric \(\alpha\)-stable processes on fractal \(d\)-sets possessing a diffusion. The stable process is subordinated to the fractal diffusion by means of the one-sided stable semigroup of index \(\alpha/2\). A Harnack inequality, an estimate of the decay rate of an \(\alpha\)-harmonic function and a Carleson estimate at the boundary are proven. On the unbounded Sierpiński gasket they obtain a boundary Harnack principle. The arguments follow \textit{K. Bogdan} [Stud. Math. 123, 43-80 (1997; Zbl 0870.31009)] with suitable adaptions to the fractal setting.