Dirichlet空间上的双线性型
Bilinear Forms on the Dirichlet Space
摘要：
设$\mathcal{D}$是圆盘上全纯函数的经典Dirichlet空间和Hilbert空间。给定一个全纯符号函数$b$，我们定义相关的汉克尔型双线性形式，最初是多项式f和g，由$T_{b}(f,g):= < fg,b >_{\mathcal{D}} $，其中我们正在寻找空间$\mathcal{D}$中的内积。我们让$T_{b}$的范数表示其范数为从$\mathcal{D}\times\mathcal{D}$到复数的双线性映射。我们称一个函数$b$在空间$\mathcal{X}$中，如果测度$dμ_{b}:=| b^{\prime}(z)| ^{2}dA$是$\mathcal{D}$的Carleson测度，范数$\mathcal{X}$由$$ \Vert b\Vert_{\mathcal{X}}:=| b(0)| +\Vert | b^{\prime}(z)| ^{2}dA\Vert_{CM(\mathcal{D})}^{1/2}. $$给出。我们的主要结果是$T_{b}$有界当且仅当$b\in\mathcal{X}$和$$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$
英文摘要：
Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by $T_{b}(f,g):= < fg,b >_{\mathcal{D}} $, where we are looking at the inner product in the space $\mathcal{D}$. We let the norm of $T_{b}$ denotes its norm as a bilinear map from $\mathcal{D}\times\mathcal{D}$ to the complex numbers. We say a function $b$ is in the space $\mathcal{X}$ if the measure $dμ_{b}:=| b^{\prime}(z)| ^{2}dA$ is a Carleson measure for $\mathcal{D}$ and norm $\mathcal{X}$ by $$ \Vert b\Vert_{\mathcal{X}}:=| b(0)| +\Vert | b^{\prime}(z)| ^{2}dA\Vert_{CM(\mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $b\in\mathcal{X}$ and $$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$
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