ar X iv : m at h / 02 06 22 4 v 1 [ m at h . C A ] 2 1 Ju n 20 02 A PRIORI L p ESTIMATES FOR SOLUTIONS OF RIEMANN – HILBERT PROBLEMS

In this paper we prove a general result establishing a priori L estimates for solutions of RiemannHilbert Problems (RHP’s) in terms of auxiliary information involving an associated “conjugate” problem (see Conjugation Lemma 1.39 below). We then use the result to obtain uniform estimates for a RHP (see Theorem 1.48) that plays a crucial role in analyzing the long-time behavior of solutions of the perturbed nonlinear Schrödinger equation on the line. Theorem 1.48 is proved by combining Conjugation Lemma 1.39 with the steepest-descent method for RHP’s introduced by the authors in [DZ1]. We do not apply the steepest-descent method directly to Theorem 1.48. Rather, as explained in the text, we proceed by rephrasing Theorem 1.48 as an equivalent inhomogeneous RHP in which the underlying objects M± (see Theorem 1.52) have appropriate analyticity properties and can be deformed around the stationary phase point much as in the manner of the classical method of stationary-phase/steepest-descent. We begin by introducing a variety of definitions and results that arise in the theory of Riemann-Hilbert Problems (RHP’s). Let Σ be an oriented contour in C and consider the associated Cauchy operator