Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators

Abstract

We give the first natural examples of Calderón-Zygmund operators in the theoryof analysis on post-critically finite self-similar fractals. This is achievedby showing that the purely imaginary Riesz and Bessel potentials on nestedfractals with $3$ or more boundary points are of this type. It follows thatthese operators are bounded on $L^{p}$, $1 < p < \infty$ and satisfy weak $1$-$1$bounds. The analysis may be extended to infinite blow-ups of these fractals,and to product spaces based on the fractal or its blow-up.