Bilinear forms on the Dirichlet space

By Nicola Arcozzi, Richard Rochberg, Eric T. Sawyer, Brett D. Wick. 2008

48 citations recorded by the source corpus.

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 $\mathcal{D}\times\mathcal{D}$ 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 $$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/cb1249a9d4658aeb6a93da97dd120e34131c13ae

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In Applied Mathematics

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Commutators in the two-weight setting (2015, background): “Several lines of investigation came out of this work: generalizations to spaces of homogeneous type, [41], multi-parameter extensions, [8,10,23,26], connections to factorization of function spaces, [1,4,6,13], div-curl lemmas, [5,22] and additional interpretations in operator theory [32, 34, 42].”