The Dirichlet-Jordan test and multidimensional extensions

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.