Off-singularity bounds and Hardy spaces for Fourier integral operators

Hassell, Andrew; Portal, Pierre; Rozendaal, Jan

doi:10.1090/tran/8090

Off-singularity bounds and Hardy spaces for Fourier integral operators
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by Andrew Hassell, Pierre Portal and Jan Rozendaal PDF

Trans. Amer. Math. Soc. 373 (2020), 5773-5832 Request permission

Abstract:

We define a scale of Hardy spaces $\mathcal {H}^{p}_{FIO}(\mathbb {R}^n)$, $p\in [1,\infty ]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$ [J. Geom. Anal. 8 (1998), pp. 629–653]. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb {R}^n$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.

References

Alex Amenta, Tent spaces over metric measure spaces under doubling and related assumptions, Operator theory in harmonic and non-commutative analysis, Oper. Theory Adv. Appl., vol. 240, Birkhäuser/Springer, Cham, 2014, pp. 1–29. MR 3134544, DOI 10.1007/978-3-319-06266-2_{1}
Pascal Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\Bbb R^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385, DOI 10.1090/memo/0871
Pascal Auscher, Steve Hofmann, and José-María Martell, Vertical versus conical square functions, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5469–5489. MR 2931335, DOI 10.1090/S0002-9947-2012-05668-6
Pascal Auscher, Christoph Kriegler, Sylvie Monniaux, and Pierre Portal, Singular integral operators on tent spaces, J. Evol. Equ. 12 (2012), no. 4, 741–765. MR 3000453, DOI 10.1007/s00028-012-0152-4
Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), no. 2, 265–316. MR 2316480, DOI 10.1007/s00028-007-0288-9
Pascal Auscher, Alan McIntosh, and Emmanuel Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), no. 1, 192–248. MR 2365673, DOI 10.1007/s12220-007-9003-x
Pascal Auscher, Sylvie Monniaux, and Pierre Portal, On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 2, 387–471. MR 3973274, DOI 10.2422/2036-2145.201601_{0}02
Pascal Auscher and Sebastian Stahlhut, Functional calculus for first order systems of Dirac type and boundary value problems, Mém. Soc. Math. Fr. (N.S.) 144 (2016), vii+164 (English, with English and French summaries). MR 3495480, DOI 10.24033/msmf.452
R. Michael Beals, $L^{p}$ boundedness of Fourier integral operators, Mem. Amer. Math. Soc. 38 (1982), no. 264, viii+57. MR 660216, DOI 10.1090/memo/0264
Emmanuel Candès and Laurent Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris 336 (2003), no. 5, 395–398 (English, with English and French summaries). MR 1979352, DOI 10.1016/S1631-073X(03)00095-5
Emmanuel J. Candès and Laurent Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math. 58 (2005), no. 11, 1472–1528. MR 2165380, DOI 10.1002/cpa.20078
Andrea Carbonaro, Alan McIntosh, and Andrew J. Morris, Local Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 23 (2013), no. 1, 106–169. MR 3010275, DOI 10.1007/s12220-011-9240-x
R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948, DOI 10.1007/BFb0058946
Elena Cordero, Fabio Nicola, and Luigi Rodino, Time-frequency analysis of Fourier integral operators, Commun. Pure Appl. Anal. 9 (2010), no. 1, 1–21. MR 2556742, DOI 10.3934/cpaa.2010.9.1
Antonio Córdoba and Charles Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979–1005. MR 507783, DOI 10.1080/03605307808820083
Sandro Coriasco and Michael Ruzhansky, Global $L^p$ continuity of Fourier integral operators, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2575–2596. MR 3165647, DOI 10.1090/S0002-9947-2014-05911-4
Xuan Thinh Duong and Lixin Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973. MR 2163867, DOI 10.1090/S0894-0347-05-00496-0
Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 320624, DOI 10.1007/BF02771772
Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
Dan-Andrei Geba and Daniel Tataru, A phase space transform adapted to the wave equation, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1065–1101. MR 2353138, DOI 10.1080/03605300600854308
Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher, Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators, J. Amer. Math. Soc. 28 (2015), no. 2, 483–529. MR 3300700, DOI 10.1090/S0894-0347-2014-00805-5
Steve Hofmann, Guozhen Lu, Dorina Mitrea, Marius Mitrea, and Lixin Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78. MR 2868142, DOI 10.1090/S0065-9266-2011-00624-6
Steve Hofmann and Svitlana Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37–116. MR 2481054, DOI 10.1007/s00208-008-0295-3
Steve Hofmann, Dorina Mitrea, Marius Mitrea, and Andrew J. Morris, $L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc. 245 (2017), no. 1159, v+108. MR 3589162, DOI 10.1090/memo/1159
Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
Lars Hörmander, The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007. Pseudo-differential operators; Reprint of the 1994 edition. MR 2304165, DOI 10.1007/978-3-540-49938-1
A. Israelsson, S. Rodriguez-Lopez, and W. Staubach, Local and global estimates for hyperbolic equations in Besov-Lipschitz and Triebel-Lizorkin spaces, Online at https://arxiv.org/abs/1802.05932, 2018.
Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653, DOI 10.1007/978-3-642-21298-7
Steven G. Krantz and Harold R. Parks, The implicit function theorem, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2013. History, theory, and applications; Reprint of the 2003 edition. MR 2977424, DOI 10.1007/978-1-4614-5981-1
André Martinez, An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, New York, 2002. MR 1872698, DOI 10.1007/978-1-4757-4495-8
Jeremy Marzuola, Jason Metcalfe, and Daniel Tataru, Wave packet parametrices for evolutions governed by PDO’s with rough symbols, Proc. Amer. Math. Soc. 136 (2008), no. 2, 597–604. MR 2358501, DOI 10.1090/S0002-9939-07-09027-2
Akihiko Miyachi, On some estimates for the wave equation in $L^{p}$ and $H^{p}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 331–354. MR 586454
Juan C. Peral, $L^{p}$ estimates for the wave equation, J. Functional Analysis 36 (1980), no. 1, 114–145. MR 568979, DOI 10.1016/0022-1236(80)90110-X
Emmanuel Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 42, Austral. Nat. Univ., Canberra, 2007, pp. 125–135. MR 2328517
Andreas Seeger, Christopher D. Sogge, and Elias M. Stein, Regularity properties of Fourier integral operators, Ann. of Math. (2) 134 (1991), no. 2, 231–251. MR 1127475, DOI 10.2307/2944346
Hart F. Smith, A Hardy space for Fourier integral operators, J. Geom. Anal. 8 (1998), no. 4, 629–653. MR 1724210, DOI 10.1007/BF02921717
Hart F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797–835 (English, with English and French summaries). MR 1644105, DOI 10.5802/aif.1640
Hart F. Smith, Spectral cluster estimates for $C^{1,1}$ metrics, Amer. J. Math. 128 (2006), no. 5, 1069–1103. MR 2262171, DOI 10.1353/ajm.2006.0041
Hart F. Smith, Propagation of singularities for rough metrics, Anal. PDE 7 (2014), no. 5, 1137–1178. MR 3265962, DOI 10.2140/apde.2014.7.1137
Christopher D. Sogge, Fourier integrals in classical analysis, 2nd ed., Cambridge Tracts in Mathematics, vol. 210, Cambridge University Press, Cambridge, 2017. MR 3645429, DOI 10.1017/9781316341186
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Daniel Tataru, The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1999) Univ. Nantes, Nantes, 1999, pp. Exp. No. XIV, 16. MR 1719002
Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), no. 2, 349–376. MR 1749052, DOI 10.1353/ajm.2000.0014
Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123 (2001), no. 3, 385–423. MR 1833146, DOI 10.1353/ajm.2001.0021
Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), no. 2, 419–442. MR 1887639, DOI 10.1090/S0894-0347-01-00375-7
Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2