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A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition

  • Mingming Cao and Qingying Xue EMAIL logo
Published/Copyright: August 5, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 2

Abstract

In this paper, we present a local Tb theorem for the non-homogeneous Littlewood–Paley gλ*-function with non-convolution type kernels and upper power bound measure μ. Actually, we show that, under the assumptions supp⁡bQ⊂Q, |∫QbQdμ|≳μ(Q) and ∥bQ∥Lp⁢(μ)p≲μ⁢(Q), the norm inequality ∥gλ*⁢(f)∥Lp⁢(μ)≲∥f∥Lp⁢(μ) holds if and only if the following testing condition holds:

supQ:cubes in ⁢ℝn1μ⁢(Q)∫Q(∫0ℓ⁢(Q)∫ℝn(tt+|x-y|)m⁢λ|θt(bQ)(y,t)|2d⁢μ⁢(y)⁢d⁢ttm+1)p/2dμ(x)<∞.

This is the first time to investigate the gλ*-function in the simultaneous presence of three attributes: local, non-homogeneous and Lp-testing condition. It is important to note that the testing condition here is an Lp type with p∈(1,2].

Keywords: Local Tb theorem; probabilistic methods
MSC 2010: 42B20; 47G10

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11471041

Award Identifier / Grant number: 11671039

Funding source: Fundamental Research Funds for the Central Universities

Award Identifier / Grant number: 2014KJJCA10

Funding statement: The authors were supported partly by NSFC (nos. 11471041 and 11671039), the Fundamental Research Funds for the Central Universities (no. 2014KJJCA10) and NCET-13-0065.

Acknowledgements

The authors want to express their sincerely thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.

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Received: 2017-2-3
Revised: 2017-6-19
Published Online: 2017-8-5
Published in Print: 2018-3-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
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https://doi.org/10.1515/forum-2017-0022
Keywords for this article
Local Tb theorem; probabilistic methods

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
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