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$L^p$ harmonic analysis on ${\rm SL}(2,{\bf R})$

About this Title

William H. Barker

Publication: Memoirs of the American Mathematical Society
Publication Year 1988: Volume 76, Number 393
ISBNs: 978-0-8218-2456-6 (print); 978-1-4704-0813-8 (online)
MathSciNet review: 946617
MSC: Primary 22E30; Secondary 22E46

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Table of Contents


  • 1. Introduction
  • 2. Notation and preliminaries
  • 3. The $L^p$ Schwartz spaces
  • 4. The principal series
  • 5. The discrete series
  • 6. Leading exponents and distributions
  • 7. Relationships between principal and discrete series matrix coefficients
  • 8. The Trombi-Varadarajan estimates for $\textrm {SL}(2, \mathbb {R})$
  • 9. The Fourier transform on $\mathcal {C}^p(G)$
  • 10. The Plancherel inversion formula
  • 11. The decomposition of $\mathcal {C}^p(G)$
  • 12. Asymptotic approximation of matrix coefficients
  • 13. Growth of asymptotic coefficients for the principal series
  • 14. Calculation of asymptotic coefficents for the discrete series
  • 15. The inverse transform
  • 16. The isomorphism theorem: Non-integral case
  • 17. The Campoli functions
  • 18. The isomorphism theorem: General case
  • 19. The zero-Schwartz space (with Henrik Schlichtkrull)
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