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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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: 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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Keywords: Normal immersions, extendable maps, holomorphic maps, Hilbert transform
Article copyright: © Copyright 1972 American Mathematical Society

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