Maximizers for the Strichartz inequality. (English) Zbl 1231.35028
Summary: We compute explicitly the best constants and, by solving some functional equations, we find all maximizers for homogeneous Strichartz estimates for the Schrödinger equation and for the wave equation in the cases when the Lebesgue exponent is an even integer.
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 39B22 | Functional equations for real functions |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35L05 | Wave equation |
Keywords:
Strichartz estimates, Schrödinger equation, wave equation, functional equations; best constantsReferences:
| [1] | Aczél, J.: Lectures on Functional Equations and Their Applications. Math. Sci. Engrg. 19, Academic Press, New York (1966) · Zbl 0139.09301 |
| [2] | Foschi, D., Klainerman, S.: Bilinear space-time estimates for homogeneous wave equations. Ann. Sci. École Norm. Sup. (4) 33 , 211-274 (2000) · Zbl 0959.35107 · doi:10.1016/S0012-9593(00)00109-9 |
| [3] | Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. 2nd ed., Springer, Berlin (1990) · Zbl 0712.35001 |
| [4] | Járai, A.: Measurability implies continuity for solutions of functional equations-even with few variables. Aequationes Math. 65 , 236-266 (2003) · Zbl 1033.39024 · doi:10.1007/s00010-003-2666-x |
| [5] | Klainerman, S., Machedon, M.: Remark on Strichartz-type inequalities. Int. Math. Res. Not. 1996 , no. 5, 201-220 · Zbl 0853.35062 · doi:10.1155/S1073792896000153 |
| [6] | Kunze, M.: On the existence of a maximizer for the Strichartz inequality. Comm. Math. Phys. 243 , 137-162 (2003) · Zbl 1060.35133 · doi:10.1007/s00220-003-0959-5 |
| [7] | Rätz, J.: On orthogonally additive mappings. Aequationes Math. 28 , 35-49 (1985) · Zbl 0569.39006 · doi:10.1007/BF02189390 |
| [8] | Strichartz, R. S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 , 705-714 (1977) · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1 |
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