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