The Leray transform: distinguished measures, symmetries and polygamma inequalities
Luke D. Edholm, Yonatan Shelah
TL;DR
The paper analyzes the Leray transform ${\bm L}$ on unbounded hypersurfaces $M_\gamma$ in ${\mathbb C}^2$, focusing on distinguished measures and projective duality to obtain explicit norm and spectral information. Central to the approach is the Leray symbol function $J(d,\gamma,k)$, which governs the $L^2$-boundedness and sub-operator norms $\|\bm{L}_k\|$, and its invariances under Hölder duality and symbol symmetries. The authors develop polygamma-based inequalities and complete monotonicity results (via Bernstein–Widder) to deduce monotonicity properties of the symbol across $k$ for various measure choices, and they deploy Euler–Maclaurin summation together with Descartes’ Rule of Signs to handle the stubborn case of the preferred symbol. They compute explicit norms for the pairing measure $\sigma$ and the Lebesgue measure $\mu_1$, and they establish that the preferred symbol $C_\nu(\gamma,k)$ is strictly increasing in $k$, yielding a closed form for the full Leray norm with the preferred measure: $\|\bm{L}\|_{L^2(M_\gamma,\nu)} = \sqrt{\dfrac{\gamma}{2\sqrt{\gamma-1}}}$. Overall, the work connects projective duality, special function inequalities, and classical summation techniques to yield precise operator norms and monotonicity phenomena for the Leray transform on a natural family of unbounded hypersurfaces.
Abstract
New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in $\mathbb{C}^2$. Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.