$L^p$-estimates of maximal function related to Schrödinger Equation in ${\mathbb{R}}^2$

Abstract

Using Guth's polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schrödinger equation in ${\mathbb{R}}^2$. The $L^p$ estimates can be used to recover the previous best known result that ${\lim }_{t\to 0}{e}^{it\Delta }f(x)=f(x)$ almost everywhere for all $f\in H^s({\mathbb{R}}^2)$ provided that $s>3/8$.

References in corpus (1)

• An improved necessary condition for the Schrödinger maximal estimate

Cited by in corpus (3)

• $L^p$-estimate of Schrödinger maximal function in higher dimensions
• Pointwise convergence of solutions to the Schrödinger equation along a class of curves
• On convergence properties for generalized Schrödinger operators along tangential curves