Sharp L^2-L^{2n} Wehrl inequalities hold for matrix coefficients of vector-valued holomorphic discrete series, with maximizers exactly the reproducing kernels.
The generalized Wehrl entropy bound in quantitative form
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abstract
Lieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided.
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Wehrl inequalities for matrix coefficients of holomorphic discrete series
Sharp L^2-L^{2n} Wehrl inequalities hold for matrix coefficients of vector-valued holomorphic discrete series, with maximizers exactly the reproducing kernels.