Oscillatory Riemann–Hilbert problems and the corona theorem

Abstract

The paper is devoted to the Riemann–Hilbert problem with matrix coefficient G∈[L ∞( R)] 2×2 having det G=1 in Hardy spaces [ H p ±] 2,1< p⩽∞, on half-planes C ± . Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann–Hilbert problem in [ H ∞ ±] 2 we study the solvability of the non-homogeneous Riemann–Hilbert problem in [ H p ±] 2,1< p<∞, 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 ∞ + are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2×2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals.