Inhomogeneous Strichartz estimates for Schrödinger's equation

Abstract

Abstract. Foschi and Vilela in their independent works ([3], [13]) showed that the range of (1/r, 1/ r) for which the inhomogeneous Strichartz estimateholds for some q, q is contained in the closed pentagon with vertices A, B, B ′ , P, P ′ except the points P, P ′ (see Figure 1). We obtain the estimate for the corner points P, P ′ . IntroductionIn this paper we consider the following Cauchy problem for the Schrödinger equation:where (x, t) ∈ R n × R, n ≥ 1. By Duhamel's principle, we have the solutionwhere e it∆ is the free Schrödinger propagator defined byThe Strichartz estimates for the solution play important roles in the study of well-posedness for nonlinear Schrödinger equations (cf. [1,12]). They actually consist of two parts, homogeneous (F = 0) and inhomogeneous (f = 0) part. The homogeneous Strichartz estimateholds if and only if (r, q) is admissible pair, that is, r, q ≥ 2, (n, r, q) = (2, ∞, 2) and n/r + 2/q = n/2 (see [11,4,9,6] and references therein). But determining the optimal range of (r, q) and ( r, q) for which the inhomogeneous Strichartz estimateholds is not completed yet when n ≥ 3. It was observed that this estimate is valid on a wider range than what is given by admissible pairs (r, q), ( r, q) (see [2], [5]).2010 Mathematics Subject Classification. Primary 35B45, 35Q40.

Inhomogeneous Strichartz estimatesin some critical cases

“…By duality, Theorem 1.1 also obtains estimates for the cases q " 8 or q " 8 which are excluded in the definition of σ-acceptable exponents used in the statements of previous results such as [5,10,7,20,12,13]. Indeed, take 0 ď θ " θ ă 1{σ in (1.15) with q " 8, in which case we obtain…”

“…In [7,20] it was shown that (1.4) holds under (1.7) and (1.8), and further assumptions which differ depending on whether d " 1, d " 2 or d ě 3; we refer the reader to these papers for the precise statement of their result (see also [12,13] for certain improvements to the range of exponents). Although (1.4) fails when pq, rq are such that (1.8) marginally fails, that is 1 q " dp 1 2 ´1 r q, the third author and Seo proved in [14] that certain weak-type estimates of the form (1.10)…”

Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials and applications

“…). Inhomogeneous Strichartz estimates with non-admissible pairs for the free Schrödinger equation have been studied by several authors [31,33,13,50,37] under suitable conditions on (p, q) (see [13,37]). The estimates (1.14) correspond to the endpoint cases for such conditions.…”

“…(4.2) was proved independently by [13] and [50]. (4.3) was settled recently by [37]. Kato-smoothing (4.4) was proved by [34].…”

Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

“…In the classical case a = 2, such an estimate was first obtained by Strichartz [44] in L q t,x (R n+1 ) norms. Since then, Strichartz's estimate has been studied by many authors [16,25,5,22,15,49,26,33,29] naturally in more general mixed norms L q t (R; L r x (R n )). (See also [12,39,34] and references therein for different related norms.)…”