Critical cyclicity in Dirichlet-type spaces on the bidisk

Abstract

Consider the Dirichlet-type spaces on the bidisc defined by $${\mathcal{D}}_{\beta }({\mathbb{D}}^2)=\{f(z_1,z_2)=\sum _{k,l}{a}_{kl}{z}_1^k{z}_2^l\in \mathcal{O}({\mathbb{D}}^2):\sum _{k,l\ge 0}|a_{kl}{|}^2(k+l+1{)}^{\beta }<+\infty \}.$$ Given ${\beta }_c\in (0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space ${\mathcal{D}}_2({\mathbb{D}}^2)$ of the bidisk and is cyclic in ${\mathcal{D}}_{\beta }({\mathbb{D}}^2)$ if and only if $\beta \le {\beta }_c.$ We also show that the critical index satisfies ${\beta }_c=2-{\dim }_H(\mathcal{Z}(f)\cap {\mathbb{T}}^2),$ where $\mathrm{di}{\mathrm{m}}_{\mathrm{H}}(\mathcal{Z}(f)\cap {\mathbb{T}}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus ${\mathbb{T}}^2.$

20 pages, comments are welcome