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

Abstract

A Titus transformation T = ⟨ α , v ⟩ T = \langle \alpha ,v\rangle is a linear operator on the vector space of C ∞ {C^\infty } mappings from the circle into the plane given by ( T f ) ( t ) = ( ⟨ α , v ⟩ f ) ( t ) = f ( t ) + α ( t ) det [ v , f ′ ( t ) ] v (Tf)(t) = (\langle \alpha ,v\rangle f)(t) = f(t) + \alpha (t)\det [v,f’(t)]v , where α \alpha is a nonnegative, C ∞ {C^\infty } function on the circle S 1 {S^1} . Let τ \tau denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of τ \tau of a degenerate curve, α 0 v 0 {\alpha _0}{v_0} , where α 0 {\alpha _0} is a C ∞ {C^\infty } function on S 1 {S^1} and v 0 {v_0} is fixed in the plane R 2 {R^2} . A C ∞ {C^\infty } mapping f : S 1 → R 2 f:{S^1} \to {R^2} is called properly extendable if there is a C ∞ {C^\infty } mapping F : D − → R 2 F:{D^ - } \to {R^2} , D the open unit disk and D − {D^ - } its closure, such that J F ≧ 0 {J_F} \geqq 0 on D , J F > 0 D,{J_F} > 0 near the boundary S 1 {S^1} of D − {D^ - } and F | s 1 = f F{|_{{s^1}}} = f . A C ∞ {C^\infty } mapping f : S 1 → R 2 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 ) = − ∫ 0 2 π k ( s ) x ( t − s ) d s 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 ) = ∫ 0 2 π cot ⁡ ( s / 2 ) x ( t − s ) d s 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.