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Some remarks on the \(L^p\)-\(L^q\) boundedness of trigonometric sums and oscillatory integrals. (English) Zbl 1099.35108

Let \(v(t,x)\) be the solution of the inhomogeneous Schrödinger equation \[ i\partial_tv-\triangle v=F(t,x),\, (t,x)\in \mathbb R\times\mathbb R^n, \] with zero initial data. The author proves that if \(0\leq \frac1{r}\leq1/2\), \(0\leq\frac1{\widetilde{r}}\leq 1/2\) and \(| 1/r-1/{\widetilde{r}}| <1/n\), then there exist some exponents \(q, \widetilde{q}\in[1,\infty]\) for which the following delayed reversed space-time estimates hold: \[ \| v\| _{L^r(\mathbb R^n;\,L^q([2,3]))}\lesssim \| F\| _{L^{\widetilde{r}'}(\mathbb R^n;\,L^{\widetilde q'}([0,1]))}, \] where \(\frac1{\widetilde{r}}+\frac1{\widetilde{r}'}=1 =\frac1{\widetilde{q}}+\frac1{\widetilde{q}'}\). To this end, the author first discusses the asymptotic behavior for the best constant in \(L^p-L^q\) estimates for trigonometric polynomials and for an integral operator which is related to the solution of inhomogeneous Schrödinger equations.
Reviewer: Yang Dachun (Kiel)

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
26D15 Inequalities for sums, series and integrals
42A05 Trigonometric polynomials, inequalities, extremal problems
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