On the regularity of inverses of singular integral operators (Q1120777)

Let G be a connected, simply connected Lie group with graded Lie algebra of homogeneous dimension D. Denote by \(L^ r_{\gamma}\) the natural left-invariant Sobolev space of order \(\gamma\) (with respect to \(L^ r(G))\). The author proves the following result. If a left-invariant linear operator S is bounded and invertible on \(L^ 2(G)\), and if its kernel is homogeneous of degree (-D) and belongs to \(L^ r_{\gamma}\) away from 0 for some \(r\in (1,\infty)\) and \(\gamma\in (0,D)\), then the kernel of \(S^{-1}\) also belongs to \(L^ r_{\gamma}\) away from 0; by Calderon-Zygmund theory, it follows that \(S^{-1}\) extends to an operator bounded on \(L^ p(G)\) for all \(p\in (1,\infty)\).