Constructions of some families of smooth Cauchy transforms

Limani, Adem  ; Malman, Bartosz

Canadian journal of mathematics, Tome 76 (2024) no. 1, pp. 319-344

Résumé

For a given Beurling–Carleson subset E of the unit circle $\mathbb {T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from $\mathbb {D}$ to $\mathbb {T}$. The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.

Détail

DOI : 10.4153/S0008414X23000081

Mots-clés : Smooth Cauchy transforms, completely nonisometric shifts, permanence principle for inner functions, de Branges–Rovnyak spaces