Singular integrals with flag kernels on homogeneous groups, I (Q446406)

Let \(G\) be a homogeneous nilpotent Lie group whose underlying space is \(\mathbb R^N\). The standard flag \(\mathcal F\) associated to the partition of \(N=a_1+\cdots+a_n\) and the decomposition \(\mathbb R^N=\mathbb R^{a_1}\oplus\cdots\oplus\mathbb R^{a_n}\), where \(a_1, \dots, a_n\) are positive integers, is defined by NEWLINE\[NEWLINE(0)\subset\mathbb R^{a_n}\subset\mathbb R^{a_{n-1}}\oplus\mathbb R^{a_n} \subset\cdots\subset\mathbb R^{a_2}\oplus\cdots\oplus\mathbb R^{a_n} \subset\mathbb R^{a_1}\oplus\cdots\oplus\mathbb R^{a_n}=\mathbb R^N.NEWLINE\]NEWLINE A flag kernel adapted to the flag \(\mathcal F\) is a distribution \(\mathcal K\in \mathcal S'(\mathbb R^N)\) satisfying some differential inequalities and cancellation conditions, where \(\mathcal S(\mathbb R^N)\) denotes the space of Schwartz functions and \(\mathcal S'(\mathbb R^N)\) its topological dual space. Let \(\mathcal A=(a_1,\dots,a_r)\) and \(\mathcal B=(b_1,\dots,b_s)\) be two partitions of \(N\) such that \(N=a_1+\cdots+a_r=b_1+\cdots+b_s\). \(\mathcal B\) is coarser than \(\mathcal A\) means that there are integers \(1=\alpha_1<\alpha_2<\cdots<\alpha_{s+1}=r+1\) such that \(b_k=\sum_{j=\alpha_k}^{\alpha_{k+1}-1}a_j\). If \(\mathcal F_{\mathcal A}\) and \(\mathcal F_{\mathcal B}\) are flags corresponding to two partitions \(\mathcal A\) and \(\mathcal B\) satisfying that \(\mathcal B\) is coarser than \(\mathcal A\), then flag \(\mathcal F_{\mathcal A}\) is finer than \(\mathcal F_{\mathcal B}\). Let \(\mathcal K\) be a flag kernel on \(\mathbb R^N\). Define the operator \(T_{\mathcal K}:\;\mathcal S(\mathbb R^N)\to L^2(\mathbb R^N)\) by setting \(T_{\mathcal K}[\phi](\mathbf{x}):=\phi\ast \mathcal K(\mathbf{x})\) for all \(\mathbf{x}\in G\) and \(\phi\in\mathcal S(\mathbb R^N)\), where the convolution is taken with respect to \(G\).NEWLINENEWLINEThe authors prove the following: Let \(\mathcal F_1\), \(\mathcal F_2\) be two standard flags on \(\mathbb R^N\) and \(\mathcal F_0\) the coarsest flag on \(\mathbb R^N\) which is finer than both \(\mathcal F_1\) and \(\mathcal F_2\). For \(j\in\{1,\,2\}\), let \(\mathcal K_j\) be a flag kernel adapted to the flag \(\mathcal F_j\). Then the composition \(T_{\mathcal K_2}\circ T_{\mathcal K_1}\) is a flag kernel adapted to the flag \(\mathcal F_0\), namely, operators of the form \(T(f)=F\ast {\mathcal K}\), with \({\mathcal K}\) being a flag kernel, form an algebra under composition. The authors also prove that if \(\mathcal K\) is a flag kernel and \(T(f):=f\ast \mathcal K\), then \(T\) is bounded on \(L^p(G)\) for \(p\in(1,\infty)\).