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

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  • 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.
    Mathematics Subject Classification: Primary: 28A80, 46F12; Secondary: 42C99, 81Q10.

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