Revista Matemática Iberoamericana

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Volume 31, Issue 1, 2015, pp. 215–244
DOI: 10.4171/RMI/831

Published online: 2015-03-05

Scalar oscillatory integrals in smooth spaces of homogeneous type

Philip T. Gressman[1]

(1) University of Pennsylvania, Philadelphia, USA

We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street (2011) on the multi-parameter Carnot–Carathéodory geometry, which endows such spaces with differential structure. The setting allows one to formulate estimates for scalar oscillatory integrals on these spaces which are uniform and respect the underlying geometry of both the space and the phase function. As a corollary we obtain a generalization of a theorem of Bruna, Nagel, and Wainger (1988) on the asymptotic behavior of scalar oscillatory integrals with smooth, convex phase of finite type.

Keywords: Oscillatory integrals, spaces of homogeneous type, stationary phase

Gressman Philip: Scalar oscillatory integrals in smooth spaces of homogeneous type. Rev. Mat. Iberoamericana 31 (2015), 215-244. doi: 10.4171/RMI/831