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Oscillating singular integral operators on compact Lie groups revisited

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Abstract
Fefferman (Acta Math 24:9-36, 1970, Theorem 2' has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian delta namely, operators of the form T-theta(-delta):=(1-delta)-n theta/4e(i(1-delta)theta 2),0 <=theta < 1. The aim of this work is to extend Fefferman's result to oscillating singular integrals on any arbitrary compact Lie group. We also consider applications to oscillating spectral multipliers of the Laplace-Beltrami operator. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.
Keywords
General Mathematics, Calderon-Zygmund operator, Weak(1,1) inequality, Oscillating singular integrals, INVARIANT OPERATORS, FOURIER MULTIPLIERS, BOUNDEDNESS

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MLA
Cardona Sanchez, Duvan, and Michael Ruzhansky. “Oscillating Singular Integral Operators on Compact Lie Groups Revisited.” MATHEMATISCHE ZEITSCHRIFT, vol. 303, no. 2, 2023, doi:10.1007/s00209-022-03175-5.
APA
Cardona Sanchez, D., & Ruzhansky, M. (2023). Oscillating singular integral operators on compact Lie groups revisited. MATHEMATISCHE ZEITSCHRIFT, 303(2). https://doi.org/10.1007/s00209-022-03175-5
Chicago author-date
Cardona Sanchez, Duvan, and Michael Ruzhansky. 2023. “Oscillating Singular Integral Operators on Compact Lie Groups Revisited.” MATHEMATISCHE ZEITSCHRIFT 303 (2). https://doi.org/10.1007/s00209-022-03175-5.
Chicago author-date (all authors)
Cardona Sanchez, Duvan, and Michael Ruzhansky. 2023. “Oscillating Singular Integral Operators on Compact Lie Groups Revisited.” MATHEMATISCHE ZEITSCHRIFT 303 (2). doi:10.1007/s00209-022-03175-5.
Vancouver
1.
Cardona Sanchez D, Ruzhansky M. Oscillating singular integral operators on compact Lie groups revisited. MATHEMATISCHE ZEITSCHRIFT. 2023;303(2).
IEEE
[1]
D. Cardona Sanchez and M. Ruzhansky, “Oscillating singular integral operators on compact Lie groups revisited,” MATHEMATISCHE ZEITSCHRIFT, vol. 303, no. 2, 2023.
@article{01GYSS4ACYEEBFJ6GKK29Y5GFT,
  abstract     = {{Fefferman (Acta Math 24:9-36, 1970, Theorem 2' has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian delta namely, operators of the form T-theta(-delta):=(1-delta)-n theta/4e(i(1-delta)theta 2),0 <=theta < 1. The aim of this work is to extend Fefferman's result to oscillating singular integrals on any arbitrary compact Lie group. We also consider applications to oscillating spectral multipliers of the Laplace-Beltrami operator. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.}},
  articleno    = {{26}},
  author       = {{Cardona Sanchez, Duvan and Ruzhansky, Michael}},
  issn         = {{0025-5874}},
  journal      = {{MATHEMATISCHE ZEITSCHRIFT}},
  keywords     = {{General Mathematics,Calderon-Zygmund operator,Weak(1,1) inequality,Oscillating singular integrals,INVARIANT OPERATORS,FOURIER MULTIPLIERS,BOUNDEDNESS}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{21}},
  title        = {{Oscillating singular integral operators on compact Lie groups revisited}},
  url          = {{http://doi.org/10.1007/s00209-022-03175-5}},
  volume       = {{303}},
  year         = {{2023}},
}

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