Abstract
AbstractWe 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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Theory and Mathematics,Analysis
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