Abstract
The wave equation, ∂ tt u=Δu, in ℝn+1, considered with initial data u(x,0)=f∈H s(ℝn) and u’(x,0)=0, has a solution which we denote by \(\frac{1}{2}(e^{it\sqrt{-\Delta}}f+e^{-it\sqrt{-\Delta}}f)\). We give almost sharp conditions under which \(\sup_{0<t<1}|e^{\pm it\sqrt{-\Delta}}f|\) and \(\sup_{t\in\mathbb{R}}|e^{\pm it\sqrt{-\Delta}}f|\) are bounded from H s(ℝn) to L q(ℝn).
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References
Bourgain, J., A remark on Schrödinger operators, Israel J. Math. 77 (1992), 1–16.
Bourgain, J., Some new estimates on oscillatory integrals, in Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser. 42, pp. 83–112, Princeton Univ. Press, Princeton, NJ, 1995.
Carbery, A., Radial Fourier multipliers and associated maximal functions, in Recent Progress in Fourier Analysis (El Escorial, 1983), North-Holland Math. Stud. 111, pp. 49–56, North-Holland, Amsterdam, 1985.
Carleson, L., Some analytic problems related to statistical mechanics, in Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, MD, 1979), Lecture Notes in Math. 779, pp. 5–45, Springer, Berlin–Heidelberg, 1980.
Cowling, M. G., Pointwise behavior of solutions to Schrödinger equations, in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math. 992, pp. 83–90, Springer, Berlin–Heidelberg, 1983.
Dahlberg, B. E. J. and Kenig, C. E., A note on the almost everywhere behavior of solutions to the Schrödinger equation, in Harmonic Analysis (Minneapolis, MN, 1981), Lecture Notes in Math. 908, pp. 205–209, Springer, Berlin–Heidelberg, 1982.
Kenig, C. E., Ponce, G. and Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33–69.
Kenig, C. E., Ponce, G. and Vega, L., Well-posedness of the initial value problem for the Korteweg–de Vries equation, J. Amer. Math. Soc. 4 (1991), 323–347.
Kenig, C. E. and Ruiz, A., A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), 239–246.
Lee, S., On pointwise convergence of the solutions to Schrödinger equations in R 2, Int. Math. Res. Not. (2006), Art. ID 32597, 21pp.
Moyua, A., Vargas, A. and Vega, L., Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not. 1996:16 (1996), 793–815.
Moyua, A., Vargas, A. and Vega, L., Restriction theorems and maximal operators related to oscillatory integrals in R 3, Duke Math. J. 96 (1999), 547–574.
Rogers, K. M., A local smoothing estimate for the Schrödinger equation, Preprint, 2007.
Rogers, K. M., Vargas, A. and Vega, L., Pointwise convergence of solutions to the nonelliptic Schrödinger equation, Indiana Univ. Math. J. 55 (2006), 1893–1906.
Rogers, K. M. and Villarroya, P., Global estimates for the Schrödinger maximal operator. Ann. Acad. Sci. Fenn. Math. 2 (2007), 425–435.
Sjölin, P., Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699–715.
Sjölin, P., Global maximal estimates for solutions to the Schrödinger equation, Studia Math. 110 (1994), 105–114.
Sjölin, P., L p maximal estimates for solutions to the Schrödinger equation, Math. Scand. 81 (1997), 35–68 (1998).
Sjölin, P., A counter-example concerning maximal estimates for solutions to equations of Schrödinger type, Indiana Univ. Math. J. 47 (1998), 593–599.
Sjölin, P., Spherical harmonics and maximal estimates for the Schrödinger equation, Ann. Acad. Sci. Fenn. Math. 30 (2005), 393–406.
Strichartz, R. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.
Tao, T., A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359–1384.
Tao, T. and Vargas, A., A bilinear approach to cone multipliers. II. Applications, Geom. Funct. Anal. 10 (2000), 216–258.
Vega, L., Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878.
Vega, L., El multiplicador de Schrödinger. La funcion maximal y los operadores de restricción, Ph.D. thesis Universidad Autónoma de Madrid, Madrid 1988.
Walther, B. G., Some L p(L ∞)- and L 2(L 2)-estimates for oscillatory Fourier transforms, in Analysis of Divergence (Orono, ME, 1997), Appl. Numer. Harmon. Anal., pp. 213–231, Birkhäuser, Boston, MA, 1999.
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Rogers, K., Villarroya, P. Sharp estimates for maximal operators associated to the wave equation. Ark Mat 46, 143–151 (2008). https://doi.org/10.1007/s11512-007-0063-8
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DOI: https://doi.org/10.1007/s11512-007-0063-8