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Summary: Global estimates of fundamental solutions for
higher-order Schr¨odinger equations
JinMyong Kim, Anton Arnold and Xiaohua Yao
Abstract. In this paper we first establish global pointwise time-space esti-
mates of the fundamental solution for Schr¨odinger equations, where the sym-
bol 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 lately already generalized to a degenerate case (cf. [4]).
Mathematics Subject Classification (2000). 42B20; 42B37; 35Q41; 35B65.
Keywords. Oscillatory integral, higher-order Schr¨odinger equation, fundamen-
tal solution estimate.
1. Introduction
In this paper we are interested in Lp
-Lq
estimates of solutions for the following
Schr¨odinger equation:
u
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