Complex Powers of the Laplacian on Affine Nested Fractals as Calder\'on-Zygmund operators

TL;DR

This paper introduces the first examples of Calderón-Zygmund operators on certain fractals, demonstrating their boundedness on L^p spaces and extending the analysis to more complex fractal structures.

Contribution

It establishes that imaginary Riesz and Bessel potentials on nested fractals are Calderón-Zygmund operators, a novel result in fractal analysis.

Findings

Operators are bounded on L^p for 1<p<∞

Operators satisfy weak 1-1 bounds

Analysis extends to fractal blow-ups and product spaces

Abstract

We give the first natural examples of Calder\'on-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with 3 or more boundary points are of this type. It follows that these 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.