Decay estimates for massive Dirac equation in a constant magnetic field

We study the deacy and Strichartz estimates for the massive Dirac Hamiltonian in a constant magnetic fields in ${\mathbb{R}}_t\times {\mathbb{R}}_x^2$: \begin{equation*} \begin{cases} i\partial_tu(t,x)-$\mathcal{D}$Au(t,x)=0, u(0,x)=f, \end{cases} \end{equation*} where ${\mathcal{D}}_A=-i\mathbf{\sigma }\cdot (\mathrm{\nabla }-i\mathbf{A}(x))+{\sigma }_{3}m$ with $m\ge 0$ being the mass and ${\sigma }_i$ being the Dirac matrices and the potential $\mathbf{A}(x)=\frac{B_0}{2}(-x_2,x_1),B_0>0$. In particular, we show the $L^1({\mathbb{R}}^2)\to L^{\mathrm{\infty }}({\mathbb{R}}^2)$ type micro-localized decay estimates, for any finite time $T>0$, there exists a constant $C_T$ such that \begin{equation*} |e^{it$\mathcal{D}${A}}\varphi(2^{-j}|$\mathcal{D}${A}|)f(x)|{[L^{\infty}($\mathbb{R}$^2)]^2} \leq C_T 2^{2j}(1+2^{j}|t|)^{-\frac12} |\varphi(2^{-j}|$\mathcal{D}${A}|)f|{[L^1{($\mathbb{R}$^2)]^2}}, \quad |t|\leq T, \end{equation*} and we further prove the local-in-time Strichartz estimates for the Dirac equations with this unbounded potential.