本学位论文在海森堡型群的背景下研究了双线性谱乘子和仿积算子的性质, 并研究了相关于~$full-Laplacian$ 算子的波方程的~$Strichartz$ 估计. 由于海森堡型群上的~$Fourier$ 变换是算子值的, 同经典欧氏空间下大不相同, 因此我们的证明中也克服了很多经典情况下没有的困难.在调和分析中, 谱乘子和仿积是两类重要的算子, 具有重大的理论意义和广泛的应用, 在许多经典问题的处理中要用到它们. 有界性是它们最重要的性质, 也是我们在本论文中着重探讨的性质.其中, 双线性谱乘子在海森堡型群上尚无定义, 如何在海森堡型群上合理的定义它们并得到非平凡的结果是我们要解决的最关键的问题之一. 我们首先在海森堡群上定义了双线性谱乘子~$T_{m}$, 并且证明了在不同的条件下, 我们定义的谱乘子满足~$T_{m}:L^{2}\times \mathcal{C}_{0}\rightarrow L^{2}$; $T_{m}:\mathcal{C}_{0}\times L^{2}\rightarrow L^{2}$; $T_{m}:L^{p}\times L^{q}\rightarrow L^{r},\frac{1}{p}+\frac{1}{q}=\frac{1}{r}, 2 ﹀

This dissertation is concerned with multilinear spectral multipliers and paraproducts and Strichartz estimates on Heisenberg-type groups. spectral multipliers and paraproducts are two important operators in Harmonic analysis. However, there are no references about multilinear spectral multipliers on Heisenberg groups. How to give a reasonable definition is very important. We first give a definition of bilinear spectral multipliers and prove its boundedness $T_{m}:L^{2}\times \mathcal{C}_{0}\rightarrow L^{2}$; $T_{m}:\mathcal{C}_{0}\times L^{2}\rightarrow L^{2}$; $T_{m}:L^{p}\times L^{q}\rightarrow L^{r},\frac{1}{p}+\frac{1}{q}=\frac{1}{r}, 2 ﹀