Asymptotically sharp bounds for the remainder for the best quadrature formulas for several classes of functions

For certain classes of functions (all functions are defined on a Jordan measurable set $G$) defined by a majorant on the modulus of continuity, we find an asymptotically sharp bound for the remainder of an optimal quadrature formula of the form
$$\int_Gf(x)\,dx\approx\sum_{\nu=1}^mc_\nu f(x^\nu)$$
When the given majorant of the modulus of continuity is $t^\alpha$ and the nonnegative function $P(x)$ is such that for any nonnegative numbera the set $\{x\in G:P(x)\le a\}$ is Jordan measurable, then we also find an asymptotically sharp bound for the remainder of an optimal quadrature formula of the form
$$\int_GP(x)f(x)\,dx\approx\sum_{\nu=1}^mc_\nu f(x^\nu)$$