Boundedness of the convolution operator in $L_p(Z^m)$ and smoothness of the symbol of the operator

Sufficient conditions for the boundedness of the convolution operator in $L_p(Z^m)$ are found. These conditions are imposed on the symbol of the operator in terms of the spaces $H_\alpha$ and $H_\beta$ (functions of bounded variation of order $\beta$). The results obtained here generalize the results of S. B. Stechkin and I. I. Hirschman [Ref. Zh. Mat.7, No. 7821 (1960)] for the one-dimensional case.