We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants ${SG}_{N}$ , $N > 3$ , proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When ${SG}_{N}$ is equipped with the standard Dirichlet form and measure $μ,$ we show there is a full $μ$ -measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$ , and that this set is not all of ${SG}_{N}$ . We also show there is a class of non-uniform measures on the usual Sierpiński Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere in sharp contrast to the case with the standard measure.

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References

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Vol. 28, No. 06

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Received 13 December 2019

Accepted 1 May 2020

Published: September 18, 2020

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HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPIŃSKI GASKETS

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