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

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.

**[1]**Donald C. Benson,*Extensions of a theorem of Loewner on integral operators*, Pacific J. Math.**9**(1959), 365–377. MR**0108691****[2]**George K. Francis,*The folded ribbon theorem. A contribution to the study of immersed circles*, Trans. Amer. Math. Soc.**141**(1969), 271–303. MR**0243542**, 10.1090/S0002-9947-1969-0243542-1**[3]**O. D. Kellogg,*Unstetigkeiten in den linearen Integralgleichungen*, Math. Ann.**58**(1904), no. 4, 441–456 (German). MR**1511245**, 10.1007/BF01449482**[4]**Charles Loewner,*A topological characterization of a class of integral operators*, Ann. of Math. (2)**49**(1948), 316–332. MR**0024487****[5]**John N. Mather,*Stability of 𝐶^{∞} mappings. II. Infinitesimal stability implies stability*, Ann. of Math. (2)**89**(1969), 254–291. MR**0259953****[6]**James R. Munkres,*Elementary differential topology*, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. MR**0198479****[7]**V. T. Norton, Jr.,*On polynomial and differential transvections of the plane*, Dissertation, University of Michigan, Ann Arbor, Mich., 1970.**[8]**C. J. Titus,*A theory of normal curves and some applications*, Pacific J. Math.**10**(1960), 1083–1096. MR**0114189****[9]**Charles J. Titus,*The combinatorial topology of analytic functions on the boundary of a disk*, Acta Math.**106**(1961), 45–64. MR**0166375****[10]**C. J. Titus,*Characerizations of the restriction of a holomorphic function to the boundary of a disk*, J. Analyse Math.**18**(1967), 351–358. MR**0212197****[11]**C. J. Titus and G. S. Young,*An extension theorem for a class of differential operators*, Michigan Math. J**6**(1959), 195–204. MR**0109345****[12]**Hassler Whitney,*On regular closed curves in the plane*, Compositio Math.**4**(1937), 276–284. MR**1556973****[13]**Gordon Thomas Whyburn,*Topological analysis*, Second, revised edition. Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N.J., 1964. MR**0165476****[14]**D. V. Widder,*The Laplace transform*, Princeton Math. Series, vol. 6, Princeton Univ. Press, Princeton, N. J., 1941. MR**3**, 232.

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Additional Information

DOI:
http://dx.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