We use methods from potential theory and harmonic analysis to show noncyclicity of polynomials on a polydisc whose zero set meets the distinguished boundary along a hypersurface. We also generalize methods used for proving cyclicity for polynomials in two variables with small zero sets to arbitrary dimension. In doing so, we show that in higher dimension, the cyclicity properties of a function do not only depend on the codimension, but also on the orientation of the zero set. Furthermore, we illustrate our results by studying a special class of polynomials. Finally, we use methods from potential theory to prove that our estimates for non-cyclicity are in fact sharp.
We study membership of rational inner functions in Dirichlet-type spaces in polydiscs. In particular, we prove a theorem relating such inclusions to $$H^p$$ H p integrability of partial derivatives of an RIF, and as a corollary we prove that all rational inner functions on $${\mathbb {D}}^n$$ D n belong to $${\mathcal {D}}_{1/n, \ldots ,1/n}({\mathbb {D}}^n)$$ D 1 / n , … , 1 / n ( D n ) . Furthermore, we show that if $$1/p \in {\mathcal {D}}_{\alpha ,\ldots ,\alpha }$$ 1 / p ∈ D α , … , α , then the RIF $$\tilde{p}/p \in {\mathcal {D}}_{\alpha +2/n,\ldots ,\alpha +2/n}$$ p ~ / p ∈ D α + 2 / n , … , α + 2 / n . Finally we illustrate how these results can be applied through several examples, and how the Łojasiewicz inequality can sometimes be applied to determine inclusion of 1/p in certain Dirichlet-type spaces.
We present two alternative proofs of Mandrekar’s theorem, which states that an invariant subspace of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition [Proc. Amer. Math. Soc. 103 (1988), pp. 145–148]. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace.
We study Clark measures associated with general two-variable rational inner functions (RIFs) on the bidisk, including those with singularities, and with general dvariable rational inner functions with no singularities. We give precise descriptions of support sets and weights for such Clark measures in terms of level sets and partial derivatives of the associated RIF. In two variables, we characterize when the associated Clark embeddings are unitary, and for generic parameter values, we relate vanishing of two-variable weights with the contact order of the associated RIF at a singularity.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.