Deviations of Riesz projections of Hill operators with singular potentials

It is shown that the deviations $P_n-P_n^0$ of Riesz projections $$P_n=\frac{1}{2\pi i}{\int }_{C_n}(z-L{)}^{-1}dz,C_n=\{\mathrm{\mid }z-n^2\mathrm{\mid }=n\},$$ of Hill operators $Ly=-y^{\mathrm{\prime }\mathrm{\prime }}+v(x)y,x\in [0,\pi ],$ with zero and $H^{-1}$ periodic potentials go to zero as $n\to \mathrm{\infty }$ even if we consider $P_n-P_n^0$ as operators from $L^1$ to $L^{\mathrm{\infty }}\mathrm{.}$ This implies that all $L^p$-norms are uniformly equivalent on the Riesz subspaces $Ran{P}_n\mathrm{.}$