Geodesic
$L^{\infty}- L^{2}$ weighted estimate for the wave equation with potential
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 2, pp. 109-135.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider a potential type perturbation of the three dimensional wave equation and we establish a dispersive estimate for the associated propagator. The main estimate is proved under the assumption that the potential $V \ge 0$ satisfies $$|V(x)| \le \frac{C}{(1+ |x|)^{2+\epsilon_{0}}},$$ where $\epsilon_{0} > 0$.
Si considera l’equazione delle onde perturbata con un potenziale in dimensione tre e si provano delle stime dispersive per il propagatore associato. La stima principale è ottenuta sotto la condizione che il potenziale $V \ge 0$ soddisfi $$|V(x)| \le \frac{C}{(1+ |x|)^{2+\epsilon_{0}}},$$ dove $\epsilon_{0} > 0$. 
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     title = {$L^{\infty}- L^{2}$ weighted estimate for the wave equation with potential},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
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Georgiev, Vladimir; Visciglia, Nicola. $L^{\infty}- L^{2}$ weighted estimate for the wave equation with potential. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 2, pp. 109-135. http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_2_a2/

[1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2(2), 1975, 151-218. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[2] P. Alsholm - G. Schmidt, Spectral and scattering theory for Schrödinger operators. Arch. Rational Mech. Anal., 40, 1970/1971, 281-311. | MR | Zbl

[3] J.A. Barcelo - A. Ruiz - L. Vega, Weighted estimates for the Helmholtz equation and some applications. J. Funct. Anal., 150(2), 1997, 356-382. | DOI | MR | Zbl

[4] M. Beals - W. Strauss, $L^{p}$ estimates for the wave equation with a potential. Comm. Partial Differential Equations, 18(7-8), 1993, 1365-1397. | DOI | MR | Zbl

[5] M. Ben-Artzi - A. Devinatz, Resolvent estimates for a sum of tensor products with applications to the spectral theory of differential operators. J. Analyse Math., 43, 1983/84, 215-250. | DOI | MR | Zbl

[6] N. Burq - F. Planchon - J. Stalker - A. Shadi Tahvildar-Zadeh, Strichartz estimates for the Wave and Schrödinger Equations with the Inverse-Square Potential. Preprint, 2002. | fulltext mini-dml | Zbl

[7] S. Cuccagna, On the wave equation with a potential. Comm. Partial Differential Equations, 25(7-8), 2000, 1549-1565. | DOI | MR | Zbl

[8] T. Ikebe, Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Rational Mech. Anal., 5, 1960, 1-34. | MR | Zbl

[9] C. Kerler, Perturbations of the Laplacian with variable coefficients in exterior domains and differentiability of the resolvent. Asymptot. Anal., 19(3-4), 1999, 209-232. | MR | Zbl

[10] F. Planchon - J. Stalker - A. Shadi Tahvildar-Zadeh, $l^{p}$ estimates for the wave equation with the inverse-square potential. Discrete Contin. Dynam. Systems, v. 9, n. 2, 2003, 427-442. | MR | Zbl

[11] M. Reed - B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York 1975. | MR | Zbl

[12] D. Robert, Autour de l’approximation semi-classique. Progress in Mathematics, 68, Birkhäuser Boston Inc., Boston, MA, 1987. | MR | Zbl

[13] J. Shatah - M. Struwe, Geometric wave equations. Courant Lecture Notes in Mathematics, 2, New York University Courant Institute of Mathematical Sciences, New York 1998. | MR | Zbl

[14] J. Sjöstrand - M. Zworski, Asymptotic distribution of resonances for convex obstacles. Acta Math., 183(2), 1999, 191-253. | DOI | MR | Zbl

[15] R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44(3), 1977, 705-714. | fulltext mini-dml | MR | Zbl

[16] B.R. Vainberg, Asymptotic methods in equations of mathematical physics. (translated from the Russian by E. Primrose). Gordon & Breach Science Publishers, New York 1989. | MR | Zbl

[17] N. Visciglia, About the Strichartz estimate and the dispersive estimate. C.R. Acad. Bulgare Sci., 55(5), 2002, 9-14. | MR | Zbl