On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

Abstract

In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger equation in the radial case. Then we apply them to the well-posedness theory for the equation i∂tu + |∇| α u = V (x, t)u, 1 < α < 2, with radialḢ γ initial data below L 2 and radial potentials V ∈ L r t L w x under the scaling-critical range α/r + n/w = α.2010 Mathematics Subject Classification. Primary: 35B45, 35A01; Secondary: 35Q40.

Generalized inhomogeneous Strichartz estimates

“…Typically, the estimates for these endpoints are ruled out and must be excluded by force. 3 For 1/q ≤ σ(1/2 − 1/p) we have the ordinary Strichartz estimates and typically, we do not have the generalized Strichartz estimates at some point for the critical decay parameter σ ′ , or not all.…”

“…Note the trivial application that the additional inhomogeneous estimates allow us to bind the weak solution to an inhomogeneous equation with zero-initial value in certain L q t L p x -norms, in which the weak solution to the homogeneous equation with non-vanishing initial value can't be bounded in general. In [3] had been considered the fractional Schrödinger equation with spherically symmetric initial data u 0 and potential V , where 1 < a < 2:…”

“…is a contraction mapping if we choose an adequate resolution space. The proof from [3] can be divided up into the following two steps:…”

“…The fractional Schrödinger equation with potential was also considered in [3] though only for spherically symmetric potentials and solutions. The main ingredient for the proof of [3, Theorem 1.2, p. 1908] were additional inhomogeneous estimates for spherically symmetric solutions.…”

“…We will recap the proof with slight modifications to see how the additional inhomogeneous estimates found after taking spherical averages from Corollary 1 allow us to drop assumptions on spherical symmetry. Due to the perturbative nature of the range of the admissible coefficients we choose not to state the integrability conditions explicitly, which was carried out in [3]. We remark that the range provided by Corollary 1 is significantly smaller than the one from [3, Corollary 1.1, p. 1907].…”