Perturbation Analysis of the Conformal Sewing Problem and Related Problems

In this dissertation, we develop two related problems in the nonlinear functional analysis. The first is the analyticity of the Cauchy singular integral in Schauder spaces which is motivated by the second problem, namely the perturbation analysis of the conformal sewing problem in Schauder and Roumieu spaces. In Chapter II, we consider the Cauchy singular integral f (t)φ0 (t) f ◦ φ(−1) (ξ) 1 1 C[φ, f ]( · ) ≡ p. v. dt = p. v. dξ 2πi ∂D φ(t) − φ(·) 2πi φ ξ − φ(·) where the oriented simple closed curve φ and the density function f are both defined on the counterclockwise oriented boundary ∂D of the plane unit disk D. Although the linear operator C[φ, ·], for a fixed φ, and the numerical computation of C[φ, f ] have been extensively studied for the last century, in view to several applications to integral equations and boundary value problems (cf. e.g. Muskhelishvili (1953) and Gakhov (1966)), the analysis of the nonlinear functional dependence of C[φ, f ] upon both its arguments seems to be a subject analyzed only more recently (see Introduction Ch. II). This new subject of research finds application in the nonlinear problems of perturbation nature which involve the Cauchy singular integral. In Chapter II we extend the analyticity result for the operator C[·, ·] of Coifman & Meyer (1983b) to a Schauder spaces setting. We assume that both φ and f belong to a Schauder space, say C∗m,α (∂D, C), of complex-valued function of class C m,α on ∂D, with m a positive natural number and α ∈ ]0, 1[. As it is well-known, under such conditions on φ and f , the function C[φ, f ](·) is also of class C m,α . By proving the unique solvability of a boundary value problem of elliptic nature in D and by applying Implicit Function Theorem to a suitable functional equation, we show the real analyticity of C[·, ·]. Then we show the complex analyticity of C[·, ·] and we compute all its differentials. This result of Lanza & Preciso (1998) will be applied in the second part of this dissertation and in another perturbation problem associated to a nonlinear integral equation (cf. Lanza & Rogosin (1997)). In Chapter III, we introduce the conformal sewing problem associated to a shift φ of ∂D, i.e. a homeomorphism of ∂D to itself. It consists in finding a pair of conformal functions (F, G) defined in D and C \ cl D, respectively, such that their continuous extensions to cl D e C \ D, Fe and G e respectively, satisfy Fe(φ(t)) = G(t) for all t ∈ ∂D. A simple normalization condition and well-known results ensure that the sewing problem associated to φ has a unique solution (F, G) and we denote by (F [·], G[·]) the pair of operators which maps φ to the trace on ∂D of such solution. The regularity properties of the operators F [φ] and G[φ] in spaces of regular functions can be used in the study of Teichmüller spaces, which constitute an important subject in geometric function theory (see Nag (1996)). Our aim is to find natural Banach spaces of regular functions where to obtain the analyticity of F [·] and G[·]. First we study the regularity of such operators in Schauder spaces C∗m,α (∂D, C), with m ≥ 1, α ∈ ]0, 1[. By using the classical integral equation approach to the sewing problem, we show that G[φ] and F [φ] = G[φ] ◦ φ(−1) belong to C∗m,α (∂D, C) when φ belongs to C∗m,α (∂D, C). In this setting, by using the analyticity of the Cauchy singular integral (cf. Ch. II) and by applying Implicit Function Theorem to a suitable integral equation, we show that G[·] extends to a complex analytic operator. Then we prove that this Schauder spaces setting is not sufficient in order to obtain an analytic extension of the operator F [·]. Indeed a natural assumption in order to have F [·] analytic, is that φ belongs to a space of real analytic functions of ∂D to C. In Chapter IV we introduce Banach spaces of real analytic functions, namely the Roumieu spaces associated to the differentiation operator. In this setting we show that G[·] and F [·] can be extended to complex analytic operators by employing the regularity results on the composition and on the inversion operator of Lanza (1994 and 1996b).