The Miura Map on the Line

We study relations between properties of the Miura map r 7→ q = B(r) = r′ + r2 and Schrödinger operators Lq = −d2/dx2 + q where r and q are real-valued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from Lloc(R) to the local Sobolev space H−1 loc (R) and the restriction of B to the Sobolev spaces Hβ(R) with β ≥ 0. For example, we prove that the image of B on Lloc(R) consists exactly of those q ∈ H−1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Korteweg-de Vries equation in H−1(R) for initial data in the range B(L2(R)) of the Miura map.