Sharp $L^p$ Convergence for Mirror-Degenerate Expansions

We analyze weighted $L^p$ convergence for the truncated reconstruction operator in the rank-one non-symmetric Heckman–Opdam setting. After localization at the mirror, the operator admits a rigid structural decomposition and reduces, up to bounded terms, to a rank-one functional. Boundedness on $L^p(w)$ is characterized by the mirror-local integrability of $w^{-\frac{1}{p-1}}$.