Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators $\mathcal L_{P,U}$ and $\mathcal L_{0,U}$ with potential $P$ summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of $P\in L_\varkappa[0,\pi]$, $\varkappa\in(1,\infty]$, equiconvergence holds for every function $\mathbf f\in L_\mu[0,\pi]$, $\mu\in[1,\infty]$, in the norm of the space $L_\nu[0,\pi]$, $\nu\in[1,\infty]$, if the indices $\varkappa,\mu$, and $\nu$ satisfy the inequality $1/\varkappa+1/\mu-1/\nu\le1$ (except for the case when $\varkappa=\nu=\infty$ and $\mu=1$). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval $[0,\pi]$ is proved for an arbitrary function $\mathbf f\in L_2[0,\pi]$.