An \((n+1)\)-fold Marcinkiewicz multiplier theorem on the Heisenberg group
In this paper the Marcinkiewicz multiplier theorem is proved on the Heisenberg group. Let \(\mathbb{H}_n= \mathbb{C}^n \times\mathbb{R}\) denote the \((2n+1)\)-dimensional Heisenberg group with the multiplication law NEWLINE\[NEWLINE(z,t) (w,s)= (z+w,t+s+2 {\mathfrak I}z \cdot\overline \omega).NEWLINE\]NEWLINE Using the coordinate \(h= (z,t)= (x+iy,t)\) for points in \(\mathbb{H}_n\), the left-invariant vector fields NEWLINE\[NEWLINEX_j= {\partial\over \partial x_j} +2y_j{\partial \over\partial t}, \quad Y_j= {\partial\over \partial y_j}- 2x_j{\partial \over\partial t},\quad j=1,2,\dots, n, \text{ and }T= {\partial\over \partial t}NEWLINE\]NEWLINE form a basis for the Lie algebra of \(\mathbb{H}_n\). The partial sub-Laplacians \({\mathcal L_1}, \dots, {\mathcal L}_n\) are given by \({\mathcal L}_j= -{1\over 4}(X_j^2+Y_j^2)\), \(j=1,\dots,n\). The operators \({\mathcal L}_1, \dots, {\mathcal L}_n\) and \(iT\) form a family of commuting self-adjoint operators and, for \(m\in L^\infty ((\mathbb{R}^+)^n \times\mathbb{R})\), we can define the joint spectral multiplier operator \(m({\mathcal L}_1, \dots, {\mathcal L}_n,iT)\), which is also given as a convolution operator with a kernel in the distribution space \({\mathcal S}'(\mathbb{H}_n)\). For \(1<p<\infty\) the author establishes the boundedness on \(L^p(\mathbb{H}_n)\) of the multiplier operators \(m({\mathcal L}_1,\dots, {\mathcal L}_n,iT)\) if \(m\) satisfies an \((n+1)\)-fold Marcinkiewicz-type condition NEWLINE\[NEWLINE\bigl|(\xi_1\partial_{\xi_1})^{i_1} \cdots(\xi_n \partial_{\xi_n})^{i_n} (\eta \partial_\eta)^j m(\xi,\eta) \bigr|\leq C_{ij}NEWLINE\]NEWLINE for all \(i=(i_1, \dots, i_n)\), \(j\leq 9n\). Furthermore, the author establishes regularity and cancellation conditions which the convolution kernels of these Marcinkiewicz-type multipliers \(m({\mathcal L}_1, \dots, {\mathcal L}_n,iT)\) satisfy.
scientific article; zbMATH DE number 1599685