The Dirichlet-Jordan test and multidimensional extensions

Abstract:

If $\mathcal {F}$ is a foliation of an open set $\Omega \subset \mathbb {R}^n$ by smooth $(n-1)$-dimensional surfaces, we define a class of functions $\mathcal {B}(\Omega ,\mathcal {F})$, supported in $\Omega$, that are, roughly speaking, smooth along $\mathcal {F}$ and of bounded variation transverse to $\mathcal {F}$. We investigate geometrical conditions on $\mathcal {F}$ that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.