A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

Abstract

We show that for an entire function $\phi$ belonging to the Fock space ${\mathcal{F}}^2({\mathbb{C}}^n)$ on the complex Euclidean space ${\mathbb{C}}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathcal{F}}^2({\mathbb{C}}^n)$ if and only if there exists a function $m\in L^{\infty }({\mathbb{R}}^n)$ such that $$\phi (z)={\int }_{{\mathbb{R}}^n}m(x)e^{-2\left(x-\frac{i}{2}z\right)\cdot \left(x-\frac{i}{2}z\right)}dx,z\in {\mathbb{C}}^n.$$ Here $d\lambda (w)={\pi }^{-n}{e}^{-{|w|}^2}dw$ is the Gaussian measure on ${\mathbb{C}}^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_{\phi }$. Moreover, we obtain the reducing subspaces of $S_{\phi }$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathcal{F}}^2(\mathbb{C})$ on the complex plane $\mathbb{C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

to appear in Advances in Mathematics