Equiconvergence of spectral decompositions of Hill-Schr\"odinger operators

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L= -d^2/dx^2 + v(x), $ $x \in [0,\pi], $ with $H_{per}^{-1} $-potential and the free operator $L^0=-d^2/dx^2, $ subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $$ \|S_N - S_N^0: L^a \to L^b \| \to 0 \quad \text{if} \;\; 1<a \leq b< \infty, \;\; 1/a - 1/b <1/2, $$ where $S_N$ and $S_N^0 $ are the $N$-th partial sums of the spectral decompositions of $L$ and $L^0.$ Moreover, if $v \in H^{-\alpha} $ with $1/2 < \alpha < 1$ and $\frac{1}{a}=(3/2)-\alpha, $ then we obtain uniform equiconvergence: $\|S_N - S_N^0: L^a \to L^\infty \| \to 0 $ as $N \to \infty. $