The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition

We prove the refined Koornwinder--Kostenko--Teschl conjecture. For the normalized weighted Jacobi function and all $n\in\mathbb N\_0$, $α,β\ge0$, and $-1\le x\le1$, $$ |g\_n^{(α,β)}(x)| \le \left[ \frac{(n+1)(n+α+β+1)} {(n+α+1)(n+β+1)} \right]^{1/4}\le1. $$ The proof combines a central contour estimate with Sturm--Sonin localization and an exact inverse moment. It also yields an $n$-uniform extreme-lobe theorem and a sharp canonical-product first-lobe principle. Applied to the discrete Laguerre evolution, the estimate gives the optimal positive-parameter decay. For $-1<α\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\|\mathrm{e}^{-\mathrm{i} tH_α}\|_{\ell^1\to\ell^\infty} =(1+t^2)^{-(1+α)/2}$. Thus the large-time decay exponent is $\min\{1,1+α\}$ for every $α>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.