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A restriction theorem for the H-type groups

Authors: Heping Liu and Yingzhan Wang
Journal: Proc. Amer. Math. Soc. 139 (2011), 2713-2720
MSC (2010): Primary 42B10, 43A65
Published electronically: March 22, 2011
MathSciNet review: 2801610
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Abstract: We prove that the restriction operator for the H-type groups is bounded from $ L^p$ to $ L^{p'}$ for $ p$ near to $ 1$ when the dimension of the center is larger than one, and the range of $ p$ depends on the dimension of the center. This is different from the Heisenberg group, on which the restriction operator is not bounded from $ L^p$ to $ L^{p'}$ unless $ p=1$.

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  • 1. M. Cowling, A. H. Dooley, A. Korányi and F. Ricci, $ H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), no. 1, 1-41. MR 1102963 (92e:22017)
  • 2. A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), no. 1, 147-153. MR 554324 (81c:58059)
  • 3. A. Kaplan and F. Ricci, Harmonic analysis on groups of Heisenberg type, Harmonic analysis, Lecture Notes in Math., 992, Springer, Berlin, 1983, 416-435. MR 729367 (85h:22017)
  • 4. H. Liu and Y. Wang, A restriction theorem for the quaternion Heisenberg group, Appl. Math. J. Chinese Univ. Series B., to appear.
  • 5. D. Müller, A restriction theorem for the Heisenberg group, Ann. of Math. (2) 131 (1990), no. 3, 567-587. MR 1053491 (91k:22021)
  • 6. H. M. Reimann, $ H$-type groups and Clifford modules, Adv. Appl. Clifford Algebras 11 (2001), no. S2, 277-287. MR 2075359 (2005j:15033)
  • 7. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, N.J., 1993. MR 1232192 (95c:42002)
  • 8. T. Tao, Some recent progress on the restriction conjecture, Fourier analysis and convexity, 217-243, Appl. Numer. Harmon. Anal., Birkhäuser Boston, 2004. MR 2087245 (2005i:42015)
  • 9. S. Thangavelu, Some restriction theorems for the Heisenberg group, Studia Math. 99 (1991), no. 1, 11-21. MR 1120736 (93e:43009)
  • 10. S. Thangavelu, Restriction theorems for the Heisenberg group, J. Reine Angew. Math. 414 (1991), 51-65. MR 1092623 (92e:22020)
  • 11. S. Thangavelu, Lectures on Hermite and Laguerre expansions, Princeton Univ. Press, 1993. MR 1215939 (94i:42001)
  • 12. S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Math., 159, Birkhäuser, 1998. MR 1633042 (99h:43001)
  • 13. Q. Yang and F. Zhu, The heat kernel on H-type groups, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1457-1464. MR 2367120 (2009h:35054)

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Additional Information

Heping Liu
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Yingzhan Wang
Affiliation: College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, People’s Republic of China

Keywords: H-type group, restriction operator, special Hermite expansion
Received by editor(s): May 12, 2010
Published electronically: March 22, 2011
Additional Notes: The authors were supported by the National Natural Science Foundation of China under Grants #10871003 and #10990012, and the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant #2007001040.
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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