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)

 
 

 

Multipliers for eigenfunction expansions of some Schrödinger operators


Author: Krzysztof Stempak
Journal: Proc. Amer. Math. Soc. 93 (1985), 477-482
MSC: Primary 43A80; Secondary 22E30, 33A75
DOI: https://doi.org/10.1090/S0002-9939-1985-0774006-9
MathSciNet review: 774006
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a graded nilpotent Lie group and let $ L$ be a positive Rockland operator on $ G$. Let $ {E_\lambda }$ denote the spectral resolution of $ L$ on $ {L^2}(G)$. A sufficient condition is given under which a function $ m$ on $ {{\mathbf{R}}^ + }$ is a $ {L^p}$-multiplier for $ L$, $ 1 < p < \infty $; that is $ {\left\Vert {\int_0^\infty {m(\lambda )d{E_\lambda }f} } \right\Vert _p} \leqslant {C_p}{\left\Vert f \right\Vert _p}$ for a constant $ {C_p}$, $ f \in {L^p}(G) \cap {L^2}(G)$. Then the same is done for an operator $ \pi (L)$, where $ \pi $ is a unitary representation of $ G$ induced from a unitary character of a normal connected subgroup $ H$ of $ G$. Hence the case of the Hermite operator $ - {d^2}/d{x^2} + {x^2}$ is covered and an $ {L^p}$-multiplier theorem for classical Hermite expansions is obtained.


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

  • [1] A. Bonami and J. L. Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphèriques, Trans. Amer. Math. Soc. 183 (1973), 223-263. MR 0338697 (49:3461)
  • [2] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., Vol. 242, Springer-Verlag, Berlin and New York, 1971. MR 0499948 (58:17690)
  • [3] W. Cupała, Certain Schrödinger operators as images of sublaplacians on nilpotent Lie groups, Studia Math. (to appear).
  • [4] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • [5] G. Folland and E. M. Stein, Hardy spaces on homogenous groups, Princeton Univ. Press, Princeton, N. J., 1982. MR 657581 (84h:43027)
  • [6] A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (to appear). MR 782662 (86g:22009)
  • [7] A. Hulanicki and J. W. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), 703-715. MR 701519 (85f:22011)
  • [8] -, Nilpotent Lie groups and summability of eigen-function expansions of Schrödinger operators, Studia Math. (to appear).
  • [9] E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Princeton Univ. Press, Princeton, N. J., 1970. MR 0252961 (40:6176)
  • [10] K. Stempak, Multipliers for eigenfunction expansions of some Schrödinger operators, 1984. MR 774006 (86i:43016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 43A80, 22E30, 33A75

Retrieve articles in all journals with MSC: 43A80, 22E30, 33A75


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0774006-9
Keywords: Multiplier, eigenfunction expansion, Rockland operator, Hermite function, homogenous type space, maximal function, Riesz means
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society