  We study whether in the setting of the Deift-Zhou nonlinear steepest descent
method one can avoid solving local parametrix problems explicitly, 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 a local parametrix
problem to a global one with constant jump matrices. Possible implications for
edge universality in random matrix theory are also discussed.
