Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The conjugation operator on $ A\sb q(G)$

Authors: Sanjiv Kumar Gupta, Shobha Madan and U. B. Tewari
Journal: Proc. Amer. Math. Soc. 121 (1994), 163-166
MSC: Primary 43A17; Secondary 42A50, 47B38
MathSciNet review: 1181167
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let G be a compact abelian group and $ \Gamma $ its dual. For $ 1 \leq q < \infty $, the space $ {A_q}(G)$ is defined as

$\displaystyle {A_q}(G) = \{ f\vert f \in {L^1}(G),\quad \hat f \in {l_q}(\Gamma )\} $

with the norm $ {\left\Vert f \right\Vert _{{A_q}}} = {\left\Vert f \right\Vert _{{L^1}}} + {\left\Vert {\hat f} \right\Vert _{{l_q}}}$. We prove: Let G be a compact, connected abelian group and P any fixed order on $ \Gamma $. If $ q > 2$ and $ \phi $ is a Young's function, then the conjugation operator H does not extend to a bounded operator from $ {A_q}(G)$ to the Orlicz space $ {L^\phi }(G)$.

References [Enhancements On Off] (What's this?)

  • [1] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. II, Grundlehren Math. Wiss., Band 152, Springer-Verlag, Berlin and New York, 1970. MR 551496 (81k:43001)
  • [2] Y. Katznelson, Introduction to harmonic analysis, Wiley, New York, 1968. MR 0248482 (40:1734)
  • [3] M. A. Krasnosel'skii and Ya. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961. MR 0126722 (23:A4016)
  • [4] R. Larsen, An introduction to the theory of multipliers, Springer-Verlag, New York and Berlin, 1971. MR 0435738 (55:8695)
  • [5] W. Rudin, Fourier analysis on groups, Interscience, New York, 1962. MR 0152834 (27:2808)
  • [6] A. Zygmund, Trigonometric series, Vol. I, Cambridge Univ. Press, New York and London, 1977. MR 0617944 (58:29731)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 43A17, 42A50, 47B38

Retrieve articles in all journals with MSC: 43A17, 42A50, 47B38

Additional Information

Keywords: Conjugation operator, Rudin-Shapiro polynomials
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society