The asymptotic $L_p$-norm of differentiated Fourier sums of functions of bounded variation

An asymptotic expression as $n\to\infty$ is found for the norms $\|S_n^{(r)}(x,f)\|_{L_q}$ ($1\le p<q<\infty$, $r=1,2,\dots$), where $S_n(x,f)$ is a Fourier sum of the $2\pi$-periodic function $f(x)$ having bounded $p$-variation. Various criteria for the continuity of a function of bounded $p$-variation are obtained as corollaries.