Global estimates of fundamental solutions for higher-order Schr\"odinger equations

TL;DR

This paper derives sharp global pointwise and Lp-Lq estimates for fundamental solutions of higher-order Schr"odinger equations with elliptic polynomial symbols, extending existing results to more general cases.

Contribution

It provides new sharp estimates for fundamental solutions of higher-order Schr"odinger equations with elliptic polynomial symbols, including degenerate cases.

Findings

Established global pointwise estimates for fundamental solutions.

Derived sharp Lp-Lq estimates for Schr"odinger solutions.

Extended known results to degenerate elliptic polynomial cases.

Abstract

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr\"odinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related Lp-Lq estimates on the Schr\"odinger solution. These estimates extend known results from the literature and are sharp. This result was latetly already generalized to a degenerate case (cf. [9]).