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

Content Provider	Semantic Scholar
Author	Farias, Antonio O.
Copyright Year	1972
Abstract	A Titus transformation T= /)(/) =/(f) + «(0 det [v,f'(t)]v, where a is a nonnegative, C° function on the circle S1. Let t denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of t of a degenerate curve, a0v0, where «o is a C function on S1 and v0 is fixed in the plane R2. A C" mapping /: S1 -*■ R2 is called properly extendable if there is a C00 mapping F: D~ -> R2, D the open unit disk and D~ its closure, such that /Fä0 on D,JF>0 near the boundary S1 of D~ and F|si =/. A CTM mapping /: S1 ->R2 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)= —¡l" k(s)x(t — s) ds. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hubert's transform of potential theory, y(t)=f2a* 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. 0. Introduction. A map/: S1 -> R2 is extendable if there is a map on the closed disk with Jacobian nonnegative throughout and positive near S1 and which agrees with/on S1. We show that any such map can be obtained by means of a finite number of growth operations (see text for precise definitions). We also make an application to a class of convolution operators introduced by C. Loewner to show they are topologically equivalent to the Hubert transform of potential theory. The proof is accomplished by approximating/from the inside by a map g which extends to a holomorphic G whose derivative has no multiple zeros, and then giving a geometric proof for such g. Presented to the Society, August 19, 1970; received by the editors December 1, 1970. AMS 1970 subject classifications. Primary 57D40, 47D05, 44A35; Secondary 30A90, 47E05, 44A15.
Starting Page	279
Ending Page	289
Page Count	11
File Format	PDF HTM / HTML
DOI	10.1090/S0002-9947-1972-0295374-6
Volume Number	167
Alternate Webpage(s)	http://www.ams.org/journals/bull/1971-77-03/S0002-9904-1971-12724-6/S0002-9904-1971-12724-6.pdf
Alternate Webpage(s)	http://www.ams.org/journals/tran/1972-167-00/S0002-9947-1972-0295374-6/S0002-9947-1972-0295374-6.pdf
Alternate Webpage(s)	https://doi.org/10.1090/S0002-9947-1972-0295374-6
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article