On multiple Fourier integrals of piecewise smooth functions with discontinuities of the second kind

It is well known that the Riesz means of eigenfunction expansions of piecewise smooth functions of order s>(n−3)/2s>(n−3)/2 converge uniformly on compacts where these functions are smooth. In 2000 L. Brandolini and L. Colzani considered eigenfunction expansions of piecewise smooth functions with discontinuities of the second kind across smooth surfaces. They showed that the Riesz means of these functions of order s>(n−3)/2s>(n−3)/2 may diverge even at certain points where these functions are smooth. Here it is argued that this effect depends on the measure of the singularity area, i.e. we consider functions with singularities across more limited areas and prove that the Riesz means of their eigenfunction expansions of order s>(n−3)/2s>(n−3)/2 converge uniformly on compacts where these functions are continuous.