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Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators

  • Wei Ding and Guozhen Lu EMAIL logo
Published/Copyright: October 26, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 6

Abstract

To study the boundedness of bi-parameter singular integral operators of non-convolution type in the Journé class, Fefferman discovered a boundedness criterion on bi-parameter Hardy spaces H p ⁢ ( ℝ n 1 × ℝ n 2 ) by considering the action of the operators on rectangle atoms. More recently, the theory of multiparameter local Hardy spaces has been developed by the authors. In this paper, we establish this type of boundedness criterion on weighted bi-parameter local Hardy spaces h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) . In comparison with the unweighted case, the uniform boundedness of rectangle atoms on weighted local bi-parameter Hardy spaces, which is crucial to establish the atomic decomposition on bi-parameter weighted local Hardy spaces, is considerably more involved. As an application, we establish the boundedness of bi-parameter pseudodifferential operators, including h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) to L ω p ⁢ ( ℝ n 1 + n 2 ) and h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) to h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) for all 0 < p ≤ 1 , which sharpens our earlier result even in the unweighted case requiring

max ⁡ { n 1 n 1 + 1 , n 2 n 2 + 1 } < p ≤ 1 .

Keywords: Weighted bi-parameter local Hardy space; Fefferman type criterion; bi-parameter pseudodifferential operator; weighted atomic decomposition
MSC 2010: 42B35; 42B30; 42B25; 42B20

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11771223

Funding source: Simons Foundation

Award Identifier / Grant number: 519099

Funding statement: The first author is partly supported by NNSF of China (grant no. 11771223). The second author is partly supported by a collaboration grant from the Simons foundation.

Acknowledgements

The authors would like to thank the referee for the very careful reading of the manuscript and many helpful comments which have helped to improve the exposition of our paper.

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Received: 2022-07-04
Revised: 2022-09-13
Published Online: 2022-10-26
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
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  5. On almost p-rational characters of p ′ -degree
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  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
https://doi.org/10.1515/forum-2022-0192
Keywords for this article
Weighted bi-parameter local Hardy space; Fefferman type criterion; bi-parameter pseudodifferential operator; weighted atomic decomposition

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
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