Boundary-value decay conjecture for Sierpinski gasket monomials

Let $r>0$, let $P_{j,k}^{(r)}$ be the degree-$j$ monomial in the $k$th family associated with the $r$-dependent Laplacian, and let $\|\cdot\|_\infty$ denote the supremum norm. Let $\lambda_2(r)$ and $\lambda_3(r)$ be the relevant Neumann eigenvalues, and let $g(r)$ be the continuous function from the ratio-existence conjecture.

Boundary-value decay conjecture. For $r\neq 1$ and $r\neq\frac{\sqrt{17}-3}{4}$,

$$\|P_{j,1}^{(r)}\|_\infty=O\left(|2\lambda_3(r)-\lambda_2(r)|^{-j}\right).$$

For $r\neq\frac{\sqrt{17}-3}{4}$,

$$\|P_{j,2}^{(r)}\|_\infty=O\left(|2\lambda_3(r)-\lambda_2(r)|^{-j}\right).$$

For all but at most finitely many values of $r$,

$$\|P_{j,3}^{(r)}\|_\infty=O\left(|g(r)|^j\right).$$

These estimates would give growth restrictions on coefficients in Taylor series on the Sierpinski gasket and help determine convergence and rearrangement behavior. They are presented as consequences of the preceding conjectural ratio statements and are not established in the supplied text.