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Lp boundedness of rough bi-parameter Fourier integral operators

  • Qing Hong , Guozhen Lu und Lu Zhang EMAIL logo
Veröffentlicht/Copyright: 19. April 2017
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 30 Heft 1

Abstract

In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form:

T⁢(f)⁢(x)=1(2⁢π)2⁢n⁢∫ℝ2⁢nei⁢φ⁢(x,ξ,η)⋅a⁢(x,ξ,η)⋅f^⁢(ξ,η)⁢𝑑ξ⁢𝑑η,

where x=(x1,x2)∈ℝn×ℝn and ξ,η∈ℝn∖{0}, a⁢(x,ξ,η)∈L∞⁢B⁢Sρm is the amplitude, and the phase function is of the form φ⁢(x,ξ,η)=φ1⁢(x1,ξ)+φ2⁢(x2,η), with φ1,φ2∈L∞⁢Φ2⁢(ℝn×ℝn∖{0}), and satisfies a certain rough non-degeneracy condition (see (2.2)).

The study of these operators are motivated by the Lp estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the Lp boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the Lp boundedness of the more general FIOs with amplitude a⁢(x,ξ,η)∈L∞⁢B⁢Sρm and non-smooth phase function φ⁢(x,ξ,η) on x satisfying a rough non-degeneracy condition.

Keywords: Bi-parameter Fourier integral operators; Seeger–Sogge–Stein decomposition; non-smooth amplitude and phase functions; non-degeneracy condition
MSC 2010: 42B20; 42B25

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11371056

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1301595

Funding statement: The first author’s research was partly supported by a grant of NNSF of China (no. 11371056), and the second and third authors’ research was partly supported by a US NSF grant (DMS-1301595). Also the third author was partly supported by a Simons Fellowship from the Simons Foundation.

Acknowledgements

We like to thank Chris Sogge for his valuable comments on our work and for his encouragement to consider possible future applications of rough bi-parameter Fourier integral operators in PDEs. We also like to thank the referee for his useful comments.

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Received: 2016-10-27
Revised: 2017-02-19
Published Online: 2017-04-19
Published in Print: 2018-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Logarithmic Sobolev inequalities for Moebius measures on spheres
  3. Delta sets for nonsymmetric numerical semigroups with embedding dimension three
  4. Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L2(ℝ+;dx/x)
  5. Remarks on Lp-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems
  6. On blocks with one modular character
  7. On the large-scale geometry of diffeomorphism groups of 1-manifolds
  8. Lp boundedness of rough bi-parameter Fourier integral operators
  9. Coclosed G2-structures inducing nilsolitons
  10. Existence of solutions for a semirelativistic Hartree equation with unbounded potentials
  11. On two questions concerning representations distinguished by the Galois involution
  12. Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains
  13. Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds
  14. The splitting of cohomology of p-groups with rank 2
  15. Star-group identities on units of group algebras: The non-torsion case
  16. Expansion for cubes in the Heisenberg group
  17. Semilinear Robin problems resonant at both zero and infinity
  18. Purely infinite simple Kumjian–Pask algebras
https://doi.org/10.1515/forum-2016-0221
Schlagwörter für diesen Artikel
Bi-parameter Fourier integral operators; Seeger–Sogge–Stein decomposition; non-smooth amplitude and phase functions; non-degeneracy condition

Artikel in diesem Heft

  1. Frontmatter
  2. Logarithmic Sobolev inequalities for Moebius measures on spheres
  3. Delta sets for nonsymmetric numerical semigroups with embedding dimension three
  4. Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L2(ℝ+;dx/x)
  5. Remarks on Lp-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems
  6. On blocks with one modular character
  7. On the large-scale geometry of diffeomorphism groups of 1-manifolds
  8. Lp boundedness of rough bi-parameter Fourier integral operators
  9. Coclosed G2-structures inducing nilsolitons
  10. Existence of solutions for a semirelativistic Hartree equation with unbounded potentials
  11. On two questions concerning representations distinguished by the Galois involution
  12. Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains
  13. Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds
  14. The splitting of cohomology of p-groups with rank 2
  15. Star-group identities on units of group algebras: The non-torsion case
  16. Expansion for cubes in the Heisenberg group
  17. Semilinear Robin problems resonant at both zero and infinity
  18. Purely infinite simple Kumjian–Pask algebras
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