| 研究実績の概要 |
New results concerning inhomogeneous Strichartz estimates for the wave equation have been obtained. After establishing that such estimates fail to be true outside the so-called acceptable region, we consider inhomogeneous Strichartz estimates on a critical line where the acceptability condition marginally fails. In such a critical case, we establish weak-type estimates. In order to achieve this, we prove that such weak-type estimates hold in an abstract setting and deduce the special case of the wave equation via an application of the dispersive estimate for the wave equation in certain Besov spaces.
In a different direction, new results have been obtained concerning smoothing estimates for velocity averages of solutions to the kinetic transport equation. In particular, in the setting of square-integrable functions, we have established a definitive result concerning smoothing estimates for velocity averages measured in hyperbolic Sobolev spaces, where the initial data is radial and where the velocity domain is either a sphere or a ball. We have also obtained extensions of these estimates to a mixed-norm setting. Our arguments rely on a variety of tools from harmonic analysis, including restriction theory of the Fourier transform.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The original proposal centred on the restriction theory of the Fourier transform. Such theory has very close connections with Strichartz estimates and our progress in the direction of inhomogeneous Strichartz estimates for the wave equation is an important contribution to this topic. In addition, we have made progress on other topics by making use of restriction theory, including smoothing estimates for velocity averages for solutions of the kinetic transport equation.
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