Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

Abstract

Abstract. We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative H p membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in H p for p ≥ 3 2 and that higher nontangential regularity of a rational inner function paradoxically reduces the H p integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for H p . Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.

The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary

“…A geometric argument then gives the original claim. For a more detailed understanding of such varieties, which has been rapidly developed in recent years, see [1,3,2,7,5].…”

Rational inner functions and their Dirichlet type norms

“…Furthermore, in [4] it is also shown that in higher dimensions it is no longer true that all partial derivatives must have the same H p integrability. Despite these limitations, in this article we prove certain results regarding regularity of the mixed partial derivative of RIFs in arbitrary dimension by proving inclusion for RIFs in certain Dirichlet type spaces; a problem more difficult than that addressed in [3] in the sense that we are dealing with both higher dimensions and higher order derivatives. Recall that D α 1 ,...,αn (D n ) is the space of holomorphic functions f (z 1 , .…”

“…In [8], Knese gives an algebraic classification of the ideal of polynomials p such that q/p lies in L 2 (T 2 ) for a fixed polynomial q, and furthermore proves that rational inner functions have non-tangential limits everywhere on the n−torus. In [3] and [5], a thorough analysis of the unimodular level sets and singular sets of RIFs gives a fairly complete understanding of the L p integrability on T 2 of the partial derivatives of RIFs in 2 variables. For instance it is shown that ∂ z 1 φ and ∂ z 2 φ have the same integrability properties.…”

“…3.6 in [1]), and the level sets are no longer guaranteed to even be continuous. This makes it difficult to find higher dimensional equivalents of many of the methods used in [3]. Furthermore, in [4] it is also shown that in higher dimensions it is no longer true that all partial derivatives must have the same H p integrability.…”

Averaged mixed Julia-Fatou type theory with applications to spectral foliation

“…They have been studied extensively in the two and several variable settings. See, e.g., [2,3,4,21,7,25].…”