Sobolev estimates for the complex Green operator on CR submanifolds of hypersurface type (Q496212)

Let \(M\) be a smooth compact pseudoconvex orientable CR submanifold of \(\mathbb C^n\) of hypersurface type, of CR-dimension \(m-1,\) which means that at each point of \(M,\) the complex tangent space has codimension one inside the real tangent space. The \(\overline\partial\)-complex induces the extrinsic \(\overline\partial_M\)-complex on \(M.\) In this paper the authors prove Sobolev estimates for the complex Green operator for \(M\) with codimension greater than one. The operators \(\overline\partial_M\) and \(\overline\partial_M^*\) have closed range at all levels. Let \(H_q\) denote the orthogonal projection of \(L^2_{(0,q)}(M) \) onto \({\text{ker}}(\overline\partial_M) \cap {\text{ker}}(\overline\partial_M^*),\) let \(S_q'\) denote the orthogonal projection of \(L^2_{(0,q)}(M) \) onto \({\text{Im}}(\overline\partial_{M,q-1})\) and \(S_q''\) the orthogonal projection of \(L^2_{(0,q)}(M) \) onto \({\text{Im}}((\overline\partial_{M,q})^*).\) In order to control commutators of vector fields with \(\overline\partial_M\) and \(\overline\partial_M^*,\) the authors suppose that a certain \(1\)-form \(\alpha\) on \(M\) is exact on the null space of the Levi form and show that for every nonnegative real number \(s,\) there is a constant \(C_s\) such that for all \(u\in L^2_{(0,q)}(M), \;0\leq q \leq m-1, \) one has \[ \begin{aligned} &\|S_q'u \|_s + \|S_q''u \|_s + \|H_qu \|_s \leq C_s \|u\|_s; \\ &\|u\|_s \leq C_s ( \| \overline\partial_M u\|_s + \| \overline\partial_M^* u\|_s + \| u \| ) , \;1\leq q \leq m-2 ; \\ &\|u\|_s \leq C_s ( \| \overline\partial_M u\|_s + \| \overline\partial_M^* u\|_s) , u \perp \mathcal H_q; \\ &\|G_qu \|_s \leq C_s \|u \|_s,\end{aligned} \] where \[ \mathcal H_q = {\text{ker}}(\overline\partial_M) \cap {\text{ker}}(\overline\partial_M^*) \] and \(G_q\) is the complex Green operator. In addition they give two classes of CR submanifolds that satisfy the assumptions of the above result.