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Endpoint estimates for a trilinear pseudo-differential operator with flag symbols

  • Jiao Chen , Liang Huang EMAIL logo and Guozhen Lu EMAIL logo
Published/Copyright: June 30, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 4

Abstract

In this paper, we establish the endpoint estimate (0<p≤1) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B⁢S1,00. The study of this operator is motivated from the Lp (1<p<∞) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L∞ and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the Lp (1<p<∞) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, Lp-estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., hp1×hp2×hp3→Lp with 1p1+1p2+1p3=1p with 0<p<∞ (see Theorem 1.6 and Theorem 1.7).

Keywords: Multi-linear flag multipliers; pseudo-differential operators; inhomogeneous Littlewood–Paley’s inequality; local Hardy spaces
MSC 2010: 42B15; 42B20; 42B25; 42B30; 42B35

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11801049

Funding source: Natural Science Foundation of Chongqing

Award Identifier / Grant number: cstc2019jcyj-msxmX0374

Award Identifier / Grant number: cstc2019jcyj-msxmX0295

Funding statement: The first two authors were supported partly by NNSF of China (Grant no. 11801049) and the Natural Science Foundation of Chongqing (Grant nos. cstc2019jcyj-msxmX0374, cstc2019jcyj-msxmX0295). The third author was partly supported by a Simons Collaboration grant from the Simons Foundation.

Acknowledgements

The authors would like to thank the referee for his/her very careful reading and comments which have improved the exposition of the paper.

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Received: 2021-01-04
Revised: 2021-05-15
Published Online: 2021-06-30
Published in Print: 2021-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Higher depth mock theta functions and q-hypergeometric series
  3. Topological and algebraic properties of universal groups for right-angled buildings
  4. On the socles of characteristically inert subgroups of Abelian p-groups
  5. Priestley duality for MV-algebras and beyond
  6. The cardinality of μM,D‐orthogonal exponentials for the planar four digits
  7. Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
  8. Ordered fields dense in their real closure and definable convex valuations
  9. Third Hankel determinants for two classes of analytic functions with real coefficients
  10. A common range problem for model spaces
  11. Generalized Ricci flow on nilpotent Lie groups
  12. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
  13. The role of the algebraic structure in Wold-type decomposition
  14. Incidences between Euclidean spaces over finite fields
  15. Cancellation in algebraic twisted sums on GL_m
https://doi.org/10.1515/forum-2021-0004
Keywords for this article
Multi-linear flag multipliers; pseudo-differential operators; inhomogeneous Littlewood–Paley’s inequality; local Hardy spaces

Articles in the same Issue

  1. Frontmatter
  2. Higher depth mock theta functions and q-hypergeometric series
  3. Topological and algebraic properties of universal groups for right-angled buildings
  4. On the socles of characteristically inert subgroups of Abelian p-groups
  5. Priestley duality for MV-algebras and beyond
  6. The cardinality of μM,D‐orthogonal exponentials for the planar four digits
  7. Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
  8. Ordered fields dense in their real closure and definable convex valuations
  9. Third Hankel determinants for two classes of analytic functions with real coefficients
  10. A common range problem for model spaces
  11. Generalized Ricci flow on nilpotent Lie groups
  12. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
  13. The role of the algebraic structure in Wold-type decomposition
  14. Incidences between Euclidean spaces over finite fields
  15. Cancellation in algebraic twisted sums on GL_m
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