Quadrature formulae for classes of functions of low smoothness

For Sobolev and Korobov spaces of functions of several variables a quadrature formula with explicitly defined coefficients and nodes is constructed. This formula is precise for trigonometric polynomials with harmonics from the corresponding step hyperbolic cross. The error of the quadrature formula in the classes $W^\alpha_p[0,1]^n$, $E^\alpha[0,1]^n$ is $o((\ln M)^\beta/M^\alpha)$, where $M$ is the number of nodes and $\beta$ is a parameter depending on the class.
The problem of the approximate calculation of multiple integrals for functions in $W^\alpha_p[0,1]^n$ is considered in the case when this class does not lie in the space of continuous functions, that is, for $\alpha\leqslant 1/p$.