Orientation-preserving mappings, a semigroup of geometric transformations, and a class of integral operators

Author:
Antonio O. Farias

Journal:
Trans. Amer. Math. Soc. **167** (1972), 279-289

MSC:
Primary 57D40; Secondary 43A80

DOI:
https://doi.org/10.1090/S0002-9947-1972-0295374-6

MathSciNet review:
0295374

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Abstract | References | Similar Articles | Additional Information

Abstract: A *Titus transformation* is a linear operator on the vector space of mappings from the circle into the plane given by , where is a nonnegative, function on the circle . Let denote the semigroup generated by finite compositions of Titus transformations. A *Titus mapping* is the image by an element of of a degenerate curve, , where is a function on and is fixed in the plane .

A mapping is called *properly extendable* if there is a mapping , *D* the open unit disk and its closure, such that on near the boundary of and . A mapping is called *normal* if it is an immersion with no triple points and all its double points are transversal.

The main result of this paper can be stated: a normal mapping is extendable if and only if it is a Titus mapping.

An application is made to a class of integral operators of the convolution type, . It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert's transform of potential theory, , which gives the relation between the real and imaginary parts of the restriction to the boundary of a function holomorphic inside the unit disk.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0295374-6

Keywords:
Normal immersions,
extendable maps,
holomorphic maps,
Hilbert transform

Article copyright:
© Copyright 1972
American Mathematical Society