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A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$

Huan Q. Bui,Evan Randles

Published 2021 in Journal of Fourier Analysis and Applications

ABSTRACT

In this article, we consider a class of functions on R, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on Z. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function P , we construct a Radon measure σP on S = {η ∈ R : P (η) = 1} which is invariant under the symmetry group of P . With this measure, we prove a generalization of the classical polarcoordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on Z and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of [20] and [21].

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