$L_{\mu }\to L_{\nu }$ equiconvergence of spectral decompositions for Dirac system with $L_{\mathrm{\varkappa }}$ potential

We consider 1d-Dirac operator ${\mathcal{L}}_{P,U}$ acting in $\mathbb H=(L_2[0,\pi])^2$ \begin{gather*} \ell(\mathbf y) = B\mathbf y + P(x)\mathbf y,\qquad B = \begin{pmatrix}-i&0\0&i\end{pmatrix},\ P(x) = \begin{pmatrix}p_1(x)&p_2(x)\ p_3(x)&p_4(x) \end{pmatrix},\qquad\mathbf y = \begin{pmatrix}y_1(x)\ y_2(x)\end{pmatrix} \end{gather*} with arbitrary regular boundary conditions $U(\mathbf{y})=0$. The functions $p_j(x)$, $1\le j\le 4$, assumed to be complex valued and summable. Any regular 1d-Dirac operator of such kind has purely discrete spectrum ${\lambda_n}{n\in\mathbb Z}$,$\lambda_n=n+O(1)$as$|n|\to\infty$\mathrm{.}Weconsiderspectraldecomposition$S{P,U}(\mathbf f)=\lim_{m\to\infty}S_{m,P,U}(\mathbf f)$ associated with operator ${\mathcal{L}}_{P,U}$: $$S_{m,P,U}(\mathbf{f})=\sum _{\mathrm{\mid }n\mathrm{\mid }\le m}[⟨\mathbf{f},{\mathbf{z}}_{2n}⟩{\mathbf{y}}_{2n}+⟨\mathbf{f},{\mathbf{z}}_{2n+1}⟩{\mathbf{y}}_{2n+1}]\mathrm{.}$$ Here ${\mathbf{y}}_n$ — eigen- and associated functions of ${\mathcal{L}}_{P,U}$ and ${\mathbf z_n}{n\in\mathbb Z}$---biorthogonalin$\mathbb H$ system. Our main result claims that for every regular 1d-Dirac operator with $p_1=p_4=0$ and $p_2,p_3\in L_{\mathrm{\varkappa }}[0,\pi ]$, $\mathrm{\varkappa }\in (1,\mathrm{\infty }]$, and for every function $\mathbf{f}(x)=\left(\begin{array}{c}{f}_1(x)\\ f_2(x)\end{array}\right)$, $f_1,f_2\in L_{\mu }[0,\pi ]$, $\mu \in [1,\mathrm{\infty }]$, the equiconvergence $$\mathrm{\parallel }{S}_{m,P,U}(\mathbf{f})-S_{m,0,U}(\mathbf{f}){\mathrm{\parallel }}_{L_{\nu }}\to 0\text{as }m\to \mathrm{\infty }$$ in the norm of the space $(L_{\nu }[0,\pi ]{)}^2$, holds provided that $1\mathrm{/}\mathrm{\varkappa }+1\mathrm{/}\mu -1\mathrm{/}\nu \le 1$ (with one exception $\mathrm{\varkappa }=\nu =\mathrm{\infty }$, $\mu =1$). In particular, in the case $\mathrm{\varkappa }=\mu =2$, $\nu =\mathrm{\infty }$ we obtain uniform pointwise equiconvergence on $[0,\pi ]$ of the series $S{m,P,U}(\mathbf f)$and$S_{m,0,U}(\mathbf f)$.