Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

Let L be the full laplacian on the Heisenberg group ${\displaystyle {ℍ}^n}$ of arbitrary dimension n. Then for ${\displaystyle f\in L^2\left({ℍ}^n\right)}$ such that ${\displaystyle {(I-L)}^{\frac{s}{2}}f\in L^2\left({ℍ}^n\right)}$, s > 3/4, for a ${\displaystyle \varphi \in C_c\left({ℍ}^n\right)}$ we have ${\displaystyle {\int }_{{ℍ}^n}|\varphi \left(x\right)|\underset{0<t\le 1}{\text{sup}}{\left|e^{(\surd -1)tL}f\left(x\right)\right|}^{2}dx\le C_{\varphi }\parallel f{\parallel }_{W^s}^2}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group ${\displaystyle {ℍ}^n}$, then for every s < 1 there exists a sequence ${\displaystyle f_n\in L^2\left({ℍ}^n\right)}$ and ${\displaystyle C_n>0}$ such that ${\displaystyle {(I-L)}^{s/2}{f}_n\in L^2\left({ℍ}^n\right)}$ and for a ${\displaystyle \varphi \in C_c\left({ℍ}^n\right)}$ we have ${\displaystyle {\int }_{{ℍ}^n}|\varphi \left(x\right)|\underset{0<t\le 1}{\text{sup}}{\left|e^{(\surd -1)t\Delta }{f}_n\left(x\right)\right|}^{2}dx\ge C_n\parallel f_n{\parallel }_{W^s}^2,\underset{n\to \infty }{lim}{C}_n=+\infty }$.