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Register a new userMultipliers between range spaces of co-analytic Toeplitz operators
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In this paper we discuss the multipliers between range spaces of co-analytic Toeplitz operators.
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In this paper we examine the multipliers from one model space to another.
Let $mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{infty}$ compact connected manifold, possibly with boundary. Fix a smoothness class $mathcal{F}=C^{infty}$, Holder $C^{k, alpha}$ or Sobolev $W^{k, p}$. The space $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ of all $mathcal{F}$ maps $V to mathcal{G}$ resp. $V to M$ is a Banach/Frechet Lie group resp. complex manifold. Let $mathcal{F}^0(V, mathcal{G})$ resp. $mathcal{F}^{0}(V, M)$ be the component of $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ containing the identity resp. constants. A map $f$ from a domain $Omega subset mathcal{F}_1(V, M)$ to $mathcal{F}_2(W, M)$ is called range decreasing if $f(x)(W) subset x(V)$, $x in Omega$. We prove that if $dim_{mathbb{R}} mathcal{G} ge 2$, then any range decreasing group homomorphism $f: mathcal{F}_1^0(V, mathcal{G}) to mathcal{F}_2(W, mathcal{G})$ is the pullback by a map $phi: W to V$. We also provide several sufficient conditions for a range decreasing holomorphic map $Omega$ $to$ $mathcal{F}_2(W, M)$ to be a pullback operator. Then we apply these results to study certain decomposition of holomorphic maps $mathcal{F}_1(V, N) supset Omega to mathcal{F}_2(W, M)$. In particular, we identify some classes of holomorphic maps $mathcal{F}_1^{0}(V, mathbb{P}^n) to mathcal{F}_2(W, mathbb{P}^m)$, including all automorphisms of $mathcal{F}^{0}(V, mathbb{P}^n)$.
For $-1<alpha<infty$, let $omega_alpha(z)=(1+alpha)(1-|z|^2)^alpha$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{mu,beta}$ between distinct weighted Bergman spaces $L_{a}^{p}(omega_{alpha})$ and $L_{a}^{q}(omega_{beta})$ when $0<pleq1$, $q=1$, $-1<alpha,beta<infty$ and $0<pleq 1<q<infty, -1<betaleqalpha<infty$, respectively. Our results can be viewed as extensions of Pau and Zhaos work in cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.
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.
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.
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