论文标题
差分运算符在Fock空间上的收缩特性
Contraction property of differential operator on Fock space
论文作者
论文摘要
在最近的论文中，\ cite {tilli}尼古拉和蒂利证明了faber-krahn不平等，对于$ p = 2 $，均显示了以下内容。如果$ f \ in \ Mathcal {f}_α^2 $是来自相应的fock空间的整个功能，则$$ \ frac {1}π\int_Ω| f（z）|^2 e^{ - π|^{ - π|^2} dx dx d x d dy \ le（1- e^e^{ - | | | | $ω$是复杂平面中的一个域，$ |ω| $是其Lebesgue度量。这种不平等是清晰的，可以达到平等。我们证明了以下尖锐的不等式$$ \int_Ω\ frac {| f^{（n）}（z）|^2e^{ - π| z |^2}}} {π^n n！ l_n（-π| z |^2）} dxdy \ le（1-e^{ - （n+1）|ω|}）\ | f \ |^2_ {2，2，π}，$ $ $ l_n $是laguerre polynomial，而$ n \ in \ in \ in \ in \ in \ in \ {0,1,2,3,3,3,3,4 \} $。对于$ n = 0 $，它与尼古拉和蒂利的结果一致。
In the recent paper, \cite{tilli} Nicola and Tilli proved the Faber-Krahn inequality, which for $p=2$, states the following. If $f\in\mathcal{F}_α^2$ is an entire function from the corresponding Fock space, then $$\frac{1}π\int_Ω |f(z)|^2 e^{-π|z|^2} dx dy \le (1-e^{-|Ω|}) \|f\|^2_{2,π}.$$ Here $Ω$ is a domain in the complex plane and $|Ω|$ is its Lebesgue measure. This inequality is sharp and equality can be attained. We prove the following sharp inequality $$\int_Ω \frac{|f^{(n)}(z)|^2e^{-π|z|^2}}{π^n n ! L_n(-π|z|^2)}dxdy \le (1-e^{-(n+1)|Ω|})\|f\|^2_{2,π},$$ 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.
