Bilinear Forms on the Dirichlet Space

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 $d\mu_{b}:=| b^{\prime}(z)| ^{2}dA$ 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 $$\Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}.$$