Uniform basis property of the system of root vectors of the Dirac operator

We study one-dimensional Dirac operator $\mathcal L$ on the segment $[0,\pi]$ with regular in the sense of Birkhoff boundary conditions $U$ and complex-valued summable potential $P=(p_{ij}(x)),$ $i,j=1,2$. We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator $\mathcal L$ assuming that boundary-value conditions $U$ and the number $\int_0^\pi(p_1(x)-p_4(x))\,dx$ are fixed and the potential $P$ takes values from the ball $B(0,R)$ of radius $R$ in the space $L_\varkappa$ for $\varkappa>1$. Moreover, we can choose the system of root functions so that it consists of eigenfunctions of the operator $\mathcal L$ except for a finite number of root vectors that can be uniformly estimated over the ball $\|P\|_\varkappa\le R$.