LAUSR

In this paper, we prove that the maximal inequality $$\begin{aligned} \big \Vert \sup _{|t|<1}|e^{it\Delta _D}f(x,y)|\big \Vert _{L^2_{\mathrm{loc}}(\Omega )}\le C\Vert f\Vert _{H^s_D(\Omega )},\quad \forall ~f\in H^s_D(\Omega ) \end{aligned}$$‖sup|t|<1|eitΔDf(x,y)|‖Lloc2(Ω)≤C‖f‖HDs(Ω),∀f∈HDs(Ω)holds for any $$s>\tfrac{1}{2}$$s>12 with $$\Omega =\{(x,y)\in \mathbb {R}^2\mid x>0\}$$Ω={(x,y)∈R2∣x>0} and $$\Delta _D=\partial _x^2+(1+x)\partial _y^2$$ΔD=∂x2+(1+x)∂y2. As a direct application, we obtain the pointwise convergence for the free Schrödinger equation $$i\partial _tu+\Delta _D u=0$$i∂tu+ΔDu=0 with initial data $$u(0)=f$$u(0)=f inside strictly convex domain.