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L 2 estimates for oscillatory integral operators and pencils of homogeneous functions

L 2 estimates for oscillatory integral operators and pencils of homogeneous functions Abstract. In this paper we study oscillatory integral operators, induced from the restricted twoplane transforms, with homogeneous polynomial phases and their L 2 decay estimates. 1991 Mathematics Subject Classi®cation: 58G15, 47G10. 1 Introduction The purpose of this paper is to prove estimates for oscillatory integral operators originating in the study of the two-plane transform. The phase functions of these oscillatory integrals are generic elements in the tensor product space of two homogeà neous polynomials of the same degree k in R m , i.e., R 2 n S k R m . The question of studying such operators was originally posed by Greenleaf and Uhlmann in [6]. Oscillatory integral operators in one variable have been treated by Phong and Stein in [12], [13]. Let MkY n be the space of a½ne k-planes in R n . The k-plane transform RkY n , de®ned by Helgason in [7], is RkY n f p f y dsyY p e MkY n y where f e C0 R n Y ds y is the normalized Lebesgue measure on the k-plane p. RkY n can be extended to E H R n by duality à hRkY n uY gi huY RkY n http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

L 2 estimates for oscillatory integral operators and pencils of homogeneous functions

Forum Mathematicum , Volume 11 (5) – Aug 1, 1999

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Publisher
de Gruyter
Copyright
Copyright (c)1999 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1999.012
Publisher site
See Article on Publisher Site

Abstract

Abstract. In this paper we study oscillatory integral operators, induced from the restricted twoplane transforms, with homogeneous polynomial phases and their L 2 decay estimates. 1991 Mathematics Subject Classi®cation: 58G15, 47G10. 1 Introduction The purpose of this paper is to prove estimates for oscillatory integral operators originating in the study of the two-plane transform. The phase functions of these oscillatory integrals are generic elements in the tensor product space of two homogeà neous polynomials of the same degree k in R m , i.e., R 2 n S k R m . The question of studying such operators was originally posed by Greenleaf and Uhlmann in [6]. Oscillatory integral operators in one variable have been treated by Phong and Stein in [12], [13]. Let MkY n be the space of a½ne k-planes in R n . The k-plane transform RkY n , de®ned by Helgason in [7], is RkY n f p f y dsyY p e MkY n y where f e C0 R n Y ds y is the normalized Lebesgue measure on the k-plane p. RkY n can be extended to E H R n by duality à hRkY n uY gi huY RkY n

Journal

Forum Mathematicumde Gruyter

Published: Aug 1, 1999

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