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- W2950877164 abstract "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):= _{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 $mathcal{D}timesmathcal{D}$ to the complex numbers. We say a function $b$ is in the space $mathcal{X}$ if the measure $dmu_{b}:=| b^{prime}(z)| ^{2}dA$ is a Carleson measure for $mathcal{D}$ and norm $mathcal{X}$ by $$ Vert bVert_{mathcal{X}}:=| b(0)| +Vert | b^{prime}(z)| ^{2}dAVert_{CM(mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $binmathcal{X}$ and $$ Vert T_{b}Vert_{mathcal{Dtimes D}}approxVert bVert_{mathcal{X}}. $$" @default.
- W2950877164 created "2019-06-27" @default.
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- W2950877164 date "2008-11-25" @default.
- W2950877164 modified "2023-09-23" @default.
- W2950877164 title "Bilinear Forms on the Dirichlet Space" @default.
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