Riemann-Hilbert theory without local parametrix problems: Applications to orthogonal polynomials

We study whether in the setting of the Deift-Zhou nonlinear steepest descent method one can avoid solving local parametrix problems, while still obtaining asymptotic results. We show that this can be done, provided an a priori estimate for the exact solution of the Riemann-Hilbert problem is known. This enables us to derive asymptotic results for orthogonal polynomials on [-1, 1] with a new class of weight functions. In these cases, the weight functions are too badly behaved to allow a reformulation of the local parametrix problem to a global one with constant jump matrices. Possible implications for edge universality in random matrix theory are also discussed. (C) 2021 The Author(s). Published by Elsevier Inc.