Hörmander's Inequality and Point Evaluations in de Branges Space
Alex Bergman
TL;DR
This paper generalizes Hörmander's inequality from real entire functions of finite exponential type to the setting of de Branges spaces, replacing cosine with the real part A_α of the Hermite–Biehler function. It proves that a real-entire f ∈ H^∞(E) achieving its maximum at ξ must dominate A_α on the interval between the adjacent zeros of B_α. The results are then applied to the norm of the embedding i_{p,E}: H^p(E) → H^∞(E) and to the structure of point-evaluation extremals, including existence, uniqueness (for p≥1), zero-separation, and orthogonality relations, in particular in the model-space setting K_Θ^p. Collectively, the work provides sharp embedding bounds, a framework for extremal analysis in de Branges/model spaces, and translations of these estimates to meromorphic inner-function model spaces.
Abstract
Let $f$ be an entire function of finite exponential type less than or equal to $σ$ which is bounded by $1$ on the real axis and satisfies $f(0) = 1$. Under these assumptions Hörmander showed that $f$ cannot decay faster than $\cos(σx)$ on the interval $(-π/σ,π/σ)$. We extend this result to the setting of de Branges spaces with cosine replaced by the real part of the associated Hermite-Biehler function. We apply this result to study the point evaluation functional and associated extremal functions in de Branges spaces (equivalently in model spaces generated by meromorphic inner functions) generalizing some recent results of Brevig, Chirre, Ortega-Cerdà, and Seip.