Combining $TT^{\ast}$ argument and bilinear interpolation, this paper obtains the Strichartz and smoothing estimates of dispersive semi-group ${\rm e}^{-{\rm i} t P(D)}$ in weighted $L^{2}$ spaces. Among other things, we recover the results in [1]. Moreover, the application of these results to the well-posedness of some equations are shown in the last section.
Jiecheng CHEN
,
Shaolei RU
,
Chenjing WU
. STRICHARTZ AND SMOOTHING ESTIMATES FOR DISPERSIVE SEMI-GROUP ${\rm e}^{-{\rm i} t P(D)}$ IN WEIGHTED ${ L}^{2}$ SPACES AND THEIR APPLICATIONS[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 401
-415
.
DOI: 10.1007/s10473-025-0208-5
[1] Koh Y, Seo I. Strichartz and smoothing estimates in weighted $ L^ 2$ spaces and their applications. Indiana Univ Math J, 2021, 70(3): 949-983
[2] Strichartz R S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math J, 1977, 44(3): 705-714
[3] Wang B X, Huo Z H, Hao C C, Guo Z H.Harmonic Analysis Method for Nonlinear Evolution Equations, I. Hackensack, NJ: World Scientific Publishing, 2011
[4] Ginibre J, Velo G. Generalized Strichartz inequalities for the wave equation. J Funct Anal, 1995, 133(1): 50-68
[5] Pecher H. Nonlinear small data scattering for the wave and Klein-Gordon equation. Math Z, 1984, 185(2): 261-270
[6] Brenner P. On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations. Math Z, 1984, 186(3): 383-391
[7] Brenner P. On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations. J Differential Equations, 1985, 56(3): 310-344
[8] Lin J L, Strauss W A. Decay and scattering of solutions of a nonlinear Schrödinger equation. J Funct Anal, 1978,30(2): 245-263
[9] Ginibre J, Velo G. Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J Math Pures Appl (9), 1985, 64(4): 363-401
[10] Barceló J A, Bennett J M, Carbery A, et al. A note on weighted estimates for the Schrödinger operator. Rev Mat Complut, 2008, 21(2): 481-488
[11] Barceló J A, Bennett J M, Carbery A, et al. Strichartz inequalities with weights in Morrey-Campanato classes. Collect Math, 2010, 61(1): 49-56
[12] Kuba M, Tao T. Endpoint Strichartz estimates. Amer J Math, 1998, 120(5): 955-980
[13] Montgomery-Smith S J. Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math J, 1998, 91(2): 393-408
[14] Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Heidelberg: Johann Ambrosius Barth, 1995
[15] Grafakos L. Classical Fourier Analysis. New York: Springer, 2008
[16] Kurtz D S. Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces. Trans Amer Math Soc, 1980, 259(1): 235-254
[17] Littman W. Fourier transforms of surface-carried measures and differentiability of surface averages. Bull Amer Math Soc, 1963, 69: 766-770
