Tauberian theorem for m-spherical transforms on the Heisenberg group

✍ Scribed by Der-Chen Chang; Wayne M. Eby

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 304 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410516

✦ Synopsis

Abstract

In this paper we prove a Tauberian type theorem for the space L $ ^1 _{\bf m} $(H~n~ ). This theorem gives sufficient conditions for a L $ ^1 _{\bf 0} $(H~n~ ) submodule J ⊂ L $ ^1 _{\bf m} $(H~n~ ) to make up all of L $ ^1 _{\bf m} $(H~n~ ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞^(H~n~ ). In connection with the Pompeiu problem, given the vanishing of integrals ∫z^m^L~g~f (z, 0) dσ (z) = 0 for all g ∈ H~n~ and i = 1, 2 for appropriate radii r~1~ and r~2~, we now have the (improved) conclusion $ {\bar {\bf Z}}^{\bf m} $ f ≡ 0, where $ {\bar {\bf Z}}^{\bf m} $ = $ \bar Z^{m_1}_1 $ · · · $ \bar Z^{m_n}_n $ and $ \bar Z_j $ form the standard basis for T^(0,1)^(H~n~ ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)