Abstract
In this paper we present LP and LP— Lq estimates for solutions of the Cauchy problem for some classes of pseudo-differential equations. First, we give estimates for operators with simple complex characteristic roots with non-negative imaginary parts. This class contains general strictly hyperbolic equations with variable coefficients. Then, we give sharp estimates for strictly hyperbolic differential operators with time dependent coefficients. The analysis is based on the corresponding LP and LP— Lq properties of Fourier integral operators with complex phase functions, which are also presented.
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References
M. Beals, LP boundedness of Fourier integrals, Mem. Amer. Math. Soc., 264 (1982).
Y. Colin de Verdière, M. Frisch, Régularité Lipschitzienne et solutions de l’équation des ondes sur une viriété Riemannienne compacte, Ann. Scient. Ecole Norm. Sup., 9 (1976), 539–565.
J.J. Duistermaat, Fourier integral operators, Birkhäuser, Boston, 1996.
L. Hörmander, L 2 estimates for Fourier integral operators with complex phase . Ark. för Matematik. 21, 283–307 (1983).
L. Hörmander, The analysis of linear partial differential operators. Vols. III-IV,Springer-Verlag, New York, Berlin, 1985.
A. Laptev, Yu. Safarov and D. Vassiliev, On global representation of Lagrangian distributions and solutions of hyperbolic equations. Comm. Pure Appl. Math. 47, 1411–1456 (1994).
W. Littman, LP-L 9 -estimates for singular integral operators,Proc. Symp. Pure and Appl. Math. A.M.S., 23 (1973) 479–481.
A. Melin and J. Sjöstrand, Fourier integral operators with complex-valued phase functions. Springer Lecture Notes. 459, 120–223 (1975).
A. Melin and J. Sjöstrand,Fourier integral operators with complex phase functions and parametrix for an interior boundary problem.Comm Partial Differential Equations 1, 313–400 (1976).
A. Miyachi, On some estimates for the wave operator in LP and HP, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 331–354.
J. Peral, LP estimates for the wave equation, J. Funct. Anal., 36 (1980), 114–145.
D.H. Phong, Regularity of Fourier integral operators, Proc. Int. Congress Math., 862–874 (1994), Zürich, Switzerland.
M. Reissig and K. Yagdjian, One application of Floquet’s theory to L p -L, estimates for hyperbolic equations with very fast oscillations, Math. Meth. Appl. Sci., 22 (1999), 937–951.
M. Reissig and K. Yagdjian, Klein-Gordon type decay rates for wave equations with time-dependent coefficients,Preprint, Univ. of Tsukuba, 1999.
M. Reissig and K. Yagdjian,L p—L q decay estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients, to appear in Math. Nachr.,214(2000).
M. Ruzhansky,Analytic Fourier integral operators,Monge-Ampère equation and holomorphic factorization, Arch. Mat.,7268–76 (1999).
M. Ruzhansky, Holomorphic factorization for the solution operators for hyperbolic equations, Int. Series of Num. Math.,130803–811 (1999).
M. Ruzhansky,On the sharpness of Seeger-Sogge-Stein orders, Hokkaido Math. J.,28357–362 (1999).
M. Ruzhansky,Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys,5599–170 (2000).
M Ruzhansky, Sharp estimates for a class of hyperbolic partial differential operators, Results in Math., to appear.
M. Ruzhansky, Regularity of Fourier integral operators with complex phase functions, to appear.
M. Ruzhansky, Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, CWI Tracts, to appear.
Yu. Safarov and D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Transl. of Math. Monographs, 155, AMS, 1997.
A. Seeger, C.D. Sogge and E.M. Stein, Regularity properties of Fourier integral operators, Ann.of Math., 134 (1991), 231–251.
C.D. Sogge, Fourier integrals in classical analysis, Cambridge University Press, 1993.
E. M. Stein, LP boundedness of certain convolution operators,Bull. Amer. Math. Soc., 77 (1971), 404–405.
E.M. Stein, Harmonic analysis, Princeton University Press, Princeton, 1993.
M. Sugimoto, On some LP -estimates for hyperbolic equations, Arkiv för Matematik, 30 (1992), 149–162.
M. Sugimoto, A priori estimates for higher order hyperbolic equations, Math. Z., 215 (1994), 519–531.
M. Sugimoto, Estimates for hyperbolic equations with non-convex characteristics, Math. Z., 222 (1996), 521–531.
M. Sugimoto, Estimates for hyperbolic equations of space dimension 3, J. Funct. Anal., 160 (1998), 382–407.
F. Treves,Introduction to pseudo-differential and Fourier integral operators,IIPlenum Press, 1982.
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Ruzhansky, M. (2001). Estimates for Pseudo-differential and Hyperbolic Differential Equations via Fourier Integrals with Complex Phases. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. ISNM International Series of Numerical Mathematics, vol 141. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8372-6_36
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DOI: https://doi.org/10.1007/978-3-0348-8372-6_36
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9538-5
Online ISBN: 978-3-0348-8372-6
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