Global estimates of fundamental solutions for higher-order Schrödinger equations

Abstract

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schrödinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L p -L q estimates on the Schrödinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).

Lp–Lqestimates for dispersive equations and related applications

“…which are exactly identical with the results in [29] based on a different method originated in [5]. If P is homogeneous and nondegenerate, then by scaling the estimates (2.8), it can be unified into the following sharp form in (t, x)-variable:…”

“…Here our proof depends on the flexible frequency decomposition method from [27,41], but needs more delicate analysis in order to obtain our decay results. In addition, it should be pointed out that our method is different from the polar coordinate transform method used in [5,2,16,17,29], since the polar coordinate transform method fails generally to degenerate cases.…”

“…Moreover, dropping homogeneity of P , based on one or another equivalently nondegenerate conditions on P , the L p -L q estimates and some related topics have also been extensively studied by several papers (see e.g. [2,5,6,10,11,16,17,27,29]). In particular, all one-dimensional cases have been covered by these papers.…”

Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

“…admits a unique solution in L ∞ (R)H k+k0 (R d ) (this is a simple application of the properties of the Fourier transform), and by a perturbative argument they also proved the global existence also for the higher oder Schrödinger equation with a bounded time-independent potential. Moreover, by following the arguments of Theorem 4.1 in [KAY12] and Lemma 4.3 in [CLM15] one obtains the following dispersive estimates and local-in-time Strichartz estimates for solutions of the linearized normal form equation (5.1).…”

Hp–Hqestimates for dispersive equations and related applications

“…Moreover, we remark that, based on one or other nondegenerate conditions on P which are all equivalent that Σ has nonzero Gaussian curvature everywhere, the L p -L q estimates and some related topics have also been extensively generalized to nonhomogeneous polynomials P (ξ) + Q(ξ ), where Q(ξ ) is any real polynomial of order less than m (see e.g. [2,5,6,9,15,23,25]). In particular, all one-dimensional cases have been covered in these papers.…”