The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space

In this paper, we are concerned with the Cauchy problem for the modified two-component Camassa–Holm system in the Besov space with data having critical regularity. The key elements in our paper are the real interpolations and logarithmic interpolation among inhomogeneous Besov space and Lemma 5.2.1 of [7] which is also called Osgood Lemma and the Fatou Lemma. The new ingredient that we introduce in this paper can be seen on pages 453–457.