On the best approximation in the metric of $L$ to certain classes of functions by Haar-system polynomials

Let $H_\omega$, $H_\omega^L$ be classes of functions $f(x)$ whose modulus of continuity $\omega(f;t)$ and, respectively, integral modulus of continuity $\omega(f;t)_L$ do not exceed a given modulus of continuity \omega(t)$, while $H_V$ is a~class of functions $f(x)$ whose variation $\mathop V\limits_0^1f$ fdoes not exceed a~given number $V>0$. Bounds are obtained for the upper limit of the best approximations in the metric of $L$ by Haar-system polynomials on the classes just introduced (on the class $H_\omega^L$ only when $\omega(t)=Kt$). These bounds are exact for class $H_V$ and, in case $\omega(t)$ is convex, also for the classes $H_\omega$ and $H\omega^L$.