Geodesic
Boundedness of the convolution operator in $L_p(Z^m)$ and smoothness of the symbol of the operator
Matematičeskie zametki, Tome 22 (1977) no. 6, pp. 873-884.

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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. 
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     author = {S. L. Edelstein},
     title = {Boundedness of the convolution operator in $L_p(Z^m)$ and smoothness of the symbol of the operator},
     journal = {Matemati\v{c}eskie zametki},
     pages = {873--884},
     publisher = {mathdoc},
     volume = {22},
     number = {6},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_6_a8/}
}
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S. L. Edelstein. Boundedness of the convolution operator in $L_p(Z^m)$ and smoothness of the symbol of the operator. Matematičeskie zametki, Tome 22 (1977) no. 6, pp. 873-884. http://geodesic.mathdoc.fr/item/MZM_1977_22_6_a8/