A restriction theorem for the H-type groups

Liu, Heping; Wang, Yingzhan

doi:10.1090/S0002-9939-2011-10907-9

A restriction theorem for the H-type groups
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by Heping Liu and Yingzhan Wang PDF

Proc. Amer. Math. Soc. 139 (2011), 2713-2720 Request permission

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 $p=1$.

References

Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), no. 1, 1–41. MR 1102963, DOI 10.1016/0001-8708(91)90060-K
Aroldo 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, DOI 10.1090/S0002-9947-1980-0554324-X
A. Kaplan and F. Ricci, Harmonic analysis on groups of Heisenberg type, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 416–435. MR 729367, DOI 10.1007/BFb0069172
H. Liu and Y. Wang, A restriction theorem for the quaternion Heisenberg group, Appl. Math. J. Chinese Univ. Series B., to appear.
Detlef Müller, A restriction theorem for the Heisenberg group, Ann. of Math. (2) 131 (1990), no. 3, 567–587. MR 1053491, DOI 10.2307/1971471
H. M. Reimann, $H$-type groups and Clifford modules, Adv. Appl. Clifford Algebras 11 (2001), no. S2, 277–287. MR 2075359, DOI 10.1007/BF03219139
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Terence Tao, Some recent progress on the restriction conjecture, Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 217–243. MR 2087245, DOI 10.1198/106186003321335099
S. Thangavelu, Some restriction theorems for the Heisenberg group, Studia Math. 99 (1991), no. 1, 11–21. MR 1120736, DOI 10.4064/sm-99-1-1-10
S. Thangavelu, Restriction theorems for the Heisenberg group, J. Reine Angew. Math. 414 (1991), 51–65. MR 1092623, DOI 10.1515/crll.1991.414.51
Sundaram Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes, vol. 42, Princeton University Press, Princeton, NJ, 1993. With a preface by Robert S. Strichartz. MR 1215939, DOI 10.1515/9780691213927
Sundaram Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Mathematics, vol. 159, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1633042, DOI 10.1007/978-1-4612-1772-5
Qiaohua Yang and Fuliu Zhu, The heat kernel on H-type groups, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1457–1464. MR 2367120, DOI 10.1090/S0002-9939-07-09257-X