THE WEIGHTED LARGE SIEVE INEQUALITIES ON LOCALLY COMPACT ABELIAN GROUPS

<正> In this paper, we have established the weighted large sieve inequalities on locally compact Abelian groups under general conditions, and particularly, we have obtained various inequalities on groups Rk, Tk and Zk. In one of these inequalities the constant implied is better than that of the existing inequality （obtained by E. Hlawka）. One is tho k-dimensional analogue of the 1-dimensional inequality with almost the best possible constants （obtained by H. L. Montgomery and R. C. Vaughan）, and another is new. Furthermore, it is shown by examples that these inequalities cannot be improved in a sense.