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Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

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Abstract

We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols a(x,η) are elements of C r* S m1,δ classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces ℌ s,pFIO (ℝn) and ℌ t,pFIO (ℝn) over the Hardy space for Fourier integral operator ℌ pFIO (ℝn). Our main result implies that for m = 0, δ =l/2 and r > n − 1, a(x, D) acts boundedly on ℌ pFIO (ℝn) for all p ∈ (1, ∞).

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References

  1. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), 209–246.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Bourdaud, Lp estimates for certain nonregular pseudodifferential operators, Comm. Partial Differential Equations 7 (1982), 1023–1033.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bownik, Anisotropic Hardy Spaces and Wavelets, American Mathematical Society, Providence, RI, 2003.

    Book  MATH  Google Scholar 

  4. R. Coifman and Y. Meyer, Au delà des pérateurs pseudo-différentiels, Astérisque 57 (1978).

  5. Z. Fan, N. Liu, J. Rozendaal and L. Song, Characterizations of the Hardy space ℌ 1FIO (ℝn) for Fourier integral operators, Studia Mathematica, in press.

  6. D. Frey and P. Portal, Lp estimates for wave equations with specific C0,1coefficients, arXiv:2010.08326 [math.AP]

  7. D.-A. Geba and D. Tataru, A phase space transform adapted to the wave equation, Comm. Partial Differential Equations 32 (2007), 1065–1101.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Genebashvili, A. Gogatishvili, V. Kokilashvili and M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Longman, Harlow, 1998.

    MATH  Google Scholar 

  9. D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27–42.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Grafakos, Modern Fourier Analysis, Springer, New York, 2014.

    Book  MATH  Google Scholar 

  11. A. Hassell, P. Portal and J. Rozendaal, Off-singularity bounds and Hardy spaces for Fourier integral operators, Trans. Amer. Math. Soc. 373 (2020), 5773–5832.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Hassell and J. Rozendaal, Lp and ℌ pFIO regularity for wave equations with rough coefficients, Part I, Pure Appl. Anal., to appear, arXiv:2010.13761 [math.AP]

  13. J. Marschall, Pseudodifferential operators with coefficients in Sobolev spaces, Trans. Amer. Math. Soc. 307 (1988), 335–361.

    MathSciNet  MATH  Google Scholar 

  14. Y. Meyer, Régularité des solutions des équations aux dérivées partielles non linéaires (d’après J.-M. Bony), in Bourbaki Seminar, Vol. 1979/80, Springer, Berlin-New York, 1981, pp. 293–302.

    Google Scholar 

  15. Y. Meyer, Remarques sur un théorème de J.-M. Bony, Rend. Circ. Mat. Palermo (2) suppl. 1 (1981), 1–20.

  16. A. Miyachi, On some estimates for the wave equation in Lpand Hp, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 331–354.

    MathSciNet  MATH  Google Scholar 

  17. J. C. Peral, Lp estimates for the wave equation, J. Funct. Anal. 36 (1980), 114–145.

    Article  MATH  Google Scholar 

  18. S. Rodríguez-López and W. Staubach, Some endpoint estimates for bilinear paraproducts and applications, J. Math. Anal. Appl. 421 (2015), 1021–1041.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Rozendaal, Characterizations of Hardy spaces for Fourier integral operators, Rev. Mat. Iberoam. 37 (2021), 1717–1745.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Seeger, C. D. Sogge and E. M. Stein, Regularity properties of Fourier integral operators, Ann. of Math. (2) 134 (1991), 231–251.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Smith, A Hardy space for Fourier integral operators, J. Geom. Anal. 8 (1998), 629–653.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Smith, A parametrix construction for wave equations with C1,1coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), 797–835.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Smith, Spectral cluster estimates for C1,1metrics, Amer. J. Math. 128 (2006), 1069–1103.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Smith, Propagation of singularities for rough metrics, Anal. PDE 7 (2014), 1137–1178.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Smith and D. Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2) 162 (2005), 291–366.

    Article  MathSciNet  MATH  Google Scholar 

  26. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  27. D. Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), 349–376.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123 (2001), 385–423.

    Article  MathSciNet  MATH  Google Scholar 

  29. D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), 419–442.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, MA, 1991.

    Book  MATH  Google Scholar 

  31. M. Taylor, Tools for PDE, American Mathematical Society, Providence, RI, 2000. potentials.

    MATH  Google Scholar 

  32. H. Triebel, Theory of Function Spaces, Birkhäuser/Springer, Basel, 2010.

    MATH  Google Scholar 

  33. F. Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), 201–222.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Andrew Hassell for many useful conversations about the article, and both Andrew Hassell and Pierre Portal for valuable advice. The author is also grateful to Dorothee Frey for a conversation regarding the use of the anisotropic multiplier theorem in Lemma 2.4, and to the anonymous referee for various helpful comments.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warsaw, Poland

    Jan Rozendaal

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  1. Jan Rozendaal
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Correspondence to Jan Rozendaal.

Additional information

This research was supported by ARC grant DP160100941, and partially supported by NCN grant UMO2017/27/B/ST1/00078. The research leading to these results has received funding from the Norwegian Financial Mechanism 2014–2021, grant 2020/37/K/ST1/02765.

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Rozendaal, J. Rough pseudodifferential operators on Hardy spaces for Fourier integral operators. JAMA 149, 135–165 (2023). https://doi.org/10.1007/s11854-022-0247-y

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  • DOI: https://doi.org/10.1007/s11854-022-0247-y

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