Contraction property of differential operator on Fock space

Abstract

In the recent paper, \cite{tilli} Nicola and Tilli proved the Faber-Krahn inequality, which for $p=2$, states the following. If $f\in {\mathcal{F}}_{\alpha }^2$ is an entire function from the corresponding Fock space, then $$\frac{1}{\pi }{\int }_{\Omega }|f(z){|}^2{e}^{-\pi |z{|}^2}dxdy\le (1-e^{-|\Omega |})\Vert f{\Vert }_{2,\pi }^2.$$ Here $\Omega$ is a domain in the complex plane and $|\Omega |$ is its Lebesgue measure. This inequality is sharp and equality can be attained. We prove the following sharp inequality $${\int }_{\Omega }\frac{|f^{(n)}(z){|}^2{e}^{-\pi |z{|}^2}}{{\pi }^{n}n!L_n(-\pi |z{|}^2)}dxdy\le (1-e^{-(n+1)|\Omega |})\Vert f{\Vert }_{2,\pi }^2,$$ where $L_n$ is Laguerre polynomial, and $n\in \{0,1,2,3,4\}$. For $n=0$ it coincides with the result of Nicola and Tilli.

14 pages