On the product theory of singular integrals
Let \(1<p<\infty\). In this nice paper, the authors establish \(L^p\)-boundedness for a class of product singular integral operators on product spaces \(\widetilde{M}=M_1\times M_2\times\cdots \times M_n\). Each factor space \(M_i\) is a smooth manifold on which the basic geometry is given by a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. In particular, the authors focus their attention on the following two specific settings: (A) Here \(M_i\) is a compact connected \(C^\infty\)-manifold. Suppose that \(k\) smooth real vector fields \(\{X_1,\,\cdots,\,X_k\}\) on \(M_i\) are given, which are of finite-type \(m\) in the sense that these vector fields together with their commutators of order \(\leq m\) span the tangent space to \(M_i\) at each point. (B) Here \(M_i\) arises as the boundary of an unbounded model polynomial domain in \({\mathbb C}^2\). Then, let \(\Omega=\{(z,\,w)\in{\mathbb C}^2| \,\text{ Im}[w]>P(z)\}\), where \(P\) is a real, subharmonic, non-harmonic polynomial of degree \(m\). Then \(M_i=\partial\Omega\) can be identified with \({\mathbb C}\times {\mathbb R}=\{(z,\,t):\;z\in{\mathbb C},\,t\in{\mathbb R}\}\). The basic \((0,1)\) Levi vector field is then \(\bar{Z}=\frac{\partial}{\partial\bar{z}}-i\frac{\partial P}{\partial\bar{z}}\frac{\partial}{\partial t}\), and let \(\bar{Z}=X_1+iX_2\). The real vector fields \(\{X_1,\,X_2\}\) and their commutators of orders \(\leq m\) span the tangent space at each point. Thus, this \(M_i\) is a special non-compact variant, with \(k=2\), of the manifolds considered in (A). The standard singular integrals on \(M_i\) are non-isotropic smoothing operators of order zero. The authors prove that the boundedness of the product operators is then a consequence of a natural Littlewood-Paley theory on \(\widetilde{M}\), which in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor. The \(L^p\) theory of product singular integrals established in this paper can be used in a number of different situations, and in particular for estimates of fundamental solutions of \(\square_b\) on certain model domains in several complex variables.
scientific article; zbMATH DE number 2110198