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

Abstract

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 $|n|\to\infty$. We consider spectral decomposition $S_{P,U}(\mathbf f)=\lim_{m\to\infty}S_{m,P,U}(\mathbf f)$ associated with operator $\mathcal L_{P,U}$: $$ S_{m,P,U}(\mathbf f) = \sum_{|n|\le m}\Big[\langle\mathbf f,\mathbf z_{2n}\rangle\mathbf y_{2n} + \langle\mathbf f,\mathbf z_{2n+1}\rangle\mathbf y_{2n+1}\Big]. $$ Here $\mathbf y_n$ --- eigen- and associated functions of $\mathcal L_{P,U}$ and $\{\mathbf z_n\}_{n\in\mathbb Z}$ --- biorthogonal in $\mathbb H$ system. Our main result claims that for every regular 1d-Dirac operator with $p_1=p_4=0$ and $p_2,\,p_3\in L_\varkappa[0,\pi]$, $\varkappa\in(1,\infty]$, and for every function $\mathbf f(x)=\begin{pmatrix}f_1(x)\\ f_2(x)\end{pmatrix}$, $f_1,\,f_2\in L_\mu[0,\pi]$, $\mu\in[1,\infty]$, the equiconvergence $$ \|S_{m,P,U}(\mathbf f)-S_{m,0,U}(\mathbf f)\|_{L_\nu}\to0\quad \text{as}\ m\to\infty $$ in the norm of the space $(L_\nu[0,\pi])^2$, holds provided that $1/\varkappa+1/\mu-1/\nu\le1$ (with one exception $\varkappa=\nu=\infty$, $\mu=1$). In particular, in the case $\varkappa=\mu=2$, $\nu=\infty$ we obtain uniform pointwise equiconvergence on $[0,\pi]$ of the series $S_{m,P,U}(\mathbf f)$ and $S_{m,0,U}(\mathbf f)$.