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

Marius Ionescu; Luke G. Rogers

doi:10.3934/cpaa.2014.13.2155

Article Contents

2014, Volume 13, Issue 6: 2155-2175. Doi: 10.3934/cpaa.2014.13.2155

This issue Previous Article Nonlinear Biharmonic Problems with Singular Potentials Next Article Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials

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

Marius Ionescu^1, and
Luke G. Rogers^2,

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

Received: May 31, 2011

Revised: June 30, 2012

Published: October 2014

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Abstract

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.

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

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