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.
Matrix valued inner functions on the bidisk have a number of natural subspaces of the Hardy space on the torus associated to them. We study their relationship to Agler decompositions, regularity up to the boundary, and restriction maps into one variable spaces. We give a complete description of the important spaces associated to matrix rational inner functions. The dimension of some of these spaces can be computed in a straightforward way, and this ends up having an application to the study of three variable rational inner functions. Examples are included to highlight the differences between the scalar and matrix cases.
We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary T 2 using level sets. We show that the unimodular level sets of a rational inner function can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of. We apply these results to obtain a detailed description of the fine numerical stability of : for instance, we show that @ @z 1 and @ @z 2 always possess the same L p-integrability on T 2 , and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.
Abstract. Let W denote a matrix A 2 weight. In this paper, we implement a scalar argument using the square function to deduce related bounds for vector-valued functions in L 2 (W ). These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on L 2 (W ). Our proof shortens the original argument by Treil and Volberg and improves the dependence on the A 2 characteristic. In particular, we prove that:, where T is either the Hilbert transform or a Haar multiplier.
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