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

TL;DR

The paper addresses sharp $L^p$ convergence for truncated reconstructions in the rank-one non-symmetric Heckman--Opdam setting (type $A_1$) with mirror degeneracy at the fixed points. It develops an orbit-based normalization of spectral indices tied to reflection orbits $\mathcal{O}_n=\{n,1-n\}$ and uses mirror localization to reduce the reconstruction kernel to a boundary, rank-one functional $\Lambda f=\int_{|y|\le \delta} f(y)\, w(y)\, dy$. Boundedness on $L^p(w)$ is shown to be completely determined by mirror-local behavior, via the criterion $\int_{|y|\le \delta} w(y)^{-\frac{1}{p-1}} dy < \infty$, with interior spectral contributions canceling in the dominant term. A concrete Example A demonstrates the criterion for a concrete mirror-local weight, illustrating that oscillation and higher-order regularity do not affect the boundedness condition. Altogether, the work provides a measure-theoretic, mirror-local criterion for sharp $L^p$ stability in this class of mirror-degenerate expansions, decoupling the global spectral sum from local weight behavior.

Abstract

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 $w^{-\frac{1}{p-1}}$.

Theorems & Definitions (11)

• Lemma 1: Reflection symmetry
• proof
• Example 2: Index orbit and normalization
• Corollary 3: Normalization on reflection orbits
• proof
• Lemma 4: Negative index reduction
• proof
• Lemma 5: Boundary collapse of the truncated kernel
• proof
• Proposition 6: Boundedness of the model functional