# American Institute of Mathematical Sciences

November  2014, 13(6): 2155-2175. doi: 10.3934/cpaa.2014.13.2155

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

 1 Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402, United States 2 Department of Mathematics, University of Connecticut, Storrs CT 06269-3009

Received  June 2011 Revised  July 2012 Published  July 2014

We give the first natural examples of Calderón-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.
Citation: Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
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##### References:
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