Extension of Strichartz's results without heat kernel estimates

Let $K$ be the fractal and assume the conditions of Theorems~ and~. Let the results of Strichartz referred to in the source be the stated spectral and fractal-analytic results, normally obtained using heat kernel estimates. Heat-kernel-free extension conjecture. Those results remain true under these conditions even without heat kernel estimates. This would extend the cited theory beyond arguments requiring heat kernel bounds; the supplied text does not state that this has been proved.