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Register a new userBoundedness criterion for integral operators on the fractional Fock-Sobolev spaces
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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 $W_H^{s,2}(mathbb R^n)$.
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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 completely characterize the boundedness of the area operators from the Bergman spaces $A^p_alpha(mathbb{B}_ n)$ to the Lebesgue spaces $L^q(mathbb{S}_ n)$ for all $0<p,q<infty$. For the case $n=1$, some partial results were previously obtained by Wu. Especially, in the case $q<p$ and $q<s$, we obtain the new characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.
In this note, we study the boundedness of integral operators $I_{g}$ and $T_{g}$ on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given.
For any real $beta$ let $H^2_beta$ be the Hardy-Sobolev space on the unit disk $D$. $H^2_beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $beta>1/2$. In this paper, we study composition operators $C_varphi$ on $H^2_beta$ for $1/2<beta<1$. Our main result is that, for a non-constant analytic function $varphi:DtoD$, the operator $C_{varphi }$ has dense range in $H_{beta }^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $varphi(D)$. It follows that if the range of $C_{varphi }$ is dense in $H_{beta }^{2}$, then $varphi $ is a weak-star generator of $H^{infty}$. Note that this conclusion is false for the classical Dirichlet space $mathfrak{D}$. We also characterize Fredholm composition operators on $H^{2}_{beta }$.
In this paper, we investigate the boundedness of Toeplitz product $T_{f}T_{g}$ and Hankel product $H_{f}^{*} H_{g}$ on Fock-Sobolev space for two polynomials $f$ and $g$ in $z,overline{z}inmathbb{C}^{n}$. As a result, the boundedness of Toeplitz operator $T_{f}$ and Hankel operator $H_{f}$ with the polynomial symbol $f$ in $z,overline{z}inmathbb{C}^{n}$ is characterized.
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