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Transactions of the American Mathematical Society

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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: A Titus transformation $ T = \langle \alpha ,v\rangle $ is a linear operator on the vector space of $ {C^\infty }$ mappings from the circle into the plane given by $ (Tf)(t) = (\langle \alpha ,v\rangle f)(t) = f(t) + \alpha (t)\det [v,f'(t)]v$, where $ \alpha$ is a nonnegative, $ {C^\infty }$ function on the circle $ {S^1}$. Let $ \tau $ denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of $ \tau $ of a degenerate curve, $ {\alpha _0}{v_0}$, where $ {\alpha _0}$ is a $ {C^\infty }$ function on $ {S^1}$ and $ {v_0}$ is fixed in the plane $ {R^2}$.

A $ {C^\infty }$ mapping $ f:{S^1} \to {R^2}$ is called properly extendable if there is a $ {C^\infty }$ mapping $ F:{D^ - } \to {R^2}$, D the open unit disk and $ {D^ - }$ its closure, such that $ {J_F} \geqq 0$ on $ D,{J_F} > 0$ near the boundary $ {S^1}$ of $ {D^ - }$ and $ F{\vert _{{s^1}}} = f$. A $ {C^\infty }$ mapping $ f:{S^1} \to {R^2}$ 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, $ y(t) = - \smallint_0^{2\pi } {k(s)x(t - s)ds} $. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert's transform of potential theory, $ y(t) = \smallint_0^{2\pi } {\cot (s/2)x(t - s)ds} $, 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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  • [1] D. C. Benson, Extensions of a theorem of Loewner on integral operations, Pacific J. Math. 9 (1959), 365-377. MR 21 #7406. MR 0108691 (21:7406)
  • [2] G. K. Francis, The folded ribbon theorem. A contribution to the study of immersed circles, Trans. Amer. Math. Soc. 141 (1969), 271-303. MR 39 #4863. MR 0243542 (39:4863)
  • [3] O. D. Kellogg, Unstetigkeit in den Integralgleichung, Math. Ann. 58 (1904), 441-456. MR 1511245
  • [4] C. Loewner, A topological characterization of a class of integral operators, Ann. of Math. (2) 49 (1948), 316-332. MR 9, 502. MR 0024487 (9:502d)
  • [5] J. N. Mather, Stability of $ {C^\infty }$ mappings. II. Infinitesimal stability implies stability, Ann. of Math. (2) 89 (1969), 254-291. MR 41 #4582. MR 0259953 (41:4582)
  • [6] J. R. Munkres, Elementary differential topology, rev. ed., Ann. of Math. Studies, no. 54, Princeton Univ. Press, Princeton, N. J., 1966. MR 33 #6637. MR 0198479 (33:6637)
  • [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 22 #5014. MR 0114189 (22:5014)
  • [9] -, The combinatorial topology of analytic functions on the boundary of a disk, Acta Math. 106 (1961), 45-64. MR 29 #3652. MR 0166375 (29:3652)
  • [10] C. J. Titus, Characterizations of the restriction of a holomorphic function to the boundary of a disk, J. Analyse Math. 18 (1967), 351-358. MR 35 #3072. MR 0212197 (35:3072)
  • [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 22 #231. MR 0109345 (22:231)
  • [12] H. Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276-284. MR 1556973
  • [13] G. T. Whyburn, Topological analysis, 2nd rev. ed., Princeton Math. Series, no. 23, Princeton Univ. Press, Princeton, N. J., 1964. MR 29 #2758. MR 0165476 (29:2758)
  • [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: 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

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