The space-time estimates for the Schrödinger equation

Abstract

In this paper, we studied the space-time estimates for the solution to the Schrödinger equation. By polynomial partitioning, induction arguments, bilinear to linear arguments and broad norm estimates, we set up several maximal estimates for the Schrödinger equation with high-frequency input data. By these maximal estimates, we obtain the sharp global space-time estimate when $n=2$ and improve the known results in the critical cases when $n\geq 3$. The maximal estimate for $n=2$ is also used to extend the results of the local space-time estimates for the solution to the Schrödinger equation.

Figures (6)

• Figure 1: $n=2$
• Figure 2: $n\geq3$
• Figure 3:
• Figure 4: $A=(\frac{13}{42},\frac{13}{42})$, $B=(\frac{5}{16},\frac{1}{4})$, $C=(\frac{7}{22},\frac{2}{11})$
• Figure 5: $P_j\rightarrow Q_j$

Theorems & Definitions (42)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3: High frequency input endpoint estimate for $n=2$
• Theorem 1.4: High frequency input estimate for $n\geq3$
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Conjecture 1.9
• Theorem 1.10