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 $${\mathbb {R}}^d$$ , 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 $${\mathbb {Z}}^d$$ . 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 $$\sigma _P$$ on $$S=\{\eta \in {\mathbb {R}}^d:P(\eta )=1\}$$ which is invariant under the symmetry group of P. With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on $${\mathbb {Z}}^d$$ 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 Randles and Saloff-Coste (J Fourier Anal Appl 21(4):754–798, 2015; Rev Mat Iberoam 33(3):1045–1121, 2017).