A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$

Abstract

In this article, we consider a class of functions on R d , 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 d . 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 σ P on S = {η ∈ R d : 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 d 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].