Oscillatory Riemann-Hilbert problems and the corona theorem

The paper is devoted to the Riemann-Hilbert problem with matrix coefficient G is an element of [L-infinity(R)](2x2) having det G=1 in Hardy spaces [H-p(+/-)](2), 1<p <=infinity, on half-planes C-+/-. Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann-Hilbert problem in [H-infinity(+)](2) we study the solvability of the non-homogeneous Riemann-Hilbert problem in [H-p(+/-)](2), 1<p<infinity, and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for G as well as explicit formulas for its factors in terms of solutions of two associated corona problems (in C+ and C-). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in H-infinity(+) are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2 x 2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals. (C) 2002 Elsevier Science (USA). All rights reserved.