No Arabic abstract
We show that for an entire function $varphi$ belonging to the Fock space ${mathscr F}^2(mathbb{C}^n)$ on the complex Euclidean space $mathbb{C}^n$, the integral operator begin{eqnarray*} S_{varphi}F(z)=int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z- bar{w}),dlambda(w), zin mathbb{C}^n, end{eqnarray*} is bounded on ${mathscr F}^2(mathbb{C}^n)$ if and only if there exists a function $min L^{infty}(mathbb{R}^n)$ such that $$ varphi(z)=int_{mathbb{R}^n} m(x)e^{-2left(x-frac{i}{2} z right)cdot left(x-frac{i}{2} z right)} dx, zin mathbb{C}^n. $$ Here $dlambda(w)= pi^{-n}e^{-leftvert wrightvert^2}dw$ 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 $S_varphi$. Moreover, we obtain the reducing subspaces of $S_{varphi}$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${mathscr F}^2(mathbb{C})$ on the complex plane ${mathbb C}$ (Integr. Equ. Oper. Theory {bf 81} (2015), 451--454).
We provide a boundedness criterion for the integral operator $S_{varphi}$ on the fractional Fock-Sobolev space $F^{s,2}(mathbb C^n)$, $sgeq 0$, where $S_{varphi}$ (introduced by Kehe Zhu) is given by begin{eqnarray*} S_{varphi}F(z):= int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z- bar{w}) dlambda(w) end{eqnarray*} with $varphi$ in the Fock space $F^2(mathbb C^n)$ and $dlambda(w): = pi^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(mathbb R^n)$.
The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle Bergman--type singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov--Sobolev space of analytic functions $B^sigma_2$ on the complex ball of $mathbb{C}^d$. In particular, we demonstrate that for any $sigma> 0$, the Carleson measures for the space are characterized by a T1 Condition. The method of proof of these results is an extension and another application of the work originated by Nazarov, Treil and the first author.
It was known to von Neumann in the 1950s that the integer lattice $mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later demonstrated by Seip and Wallsten that a sequence of points $Gamma$ that is uniformly close to the integer lattice is still a uniqueness set. We show in this paper that the uniqueness sets for the Fock space are preserved under much more general perturbations.
We obtain sufficient conditions for a densely defined operator on the Fock space to be bounded or compact. Under the boundedness condition we then characterize the compactness of the operator in terms of its Berezin transform.
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.