RESTRICTIONS FROM R(N) TO Z(N) OF WEAK TYPE (1, 1) MULTIPLIERS

Suppose that {phi(j)}j=1infinity is a sequence of weak type (1, 1) mul tipliers for L1(R(n)) such that for each j, phi(j) is continuous at ev ery point of Z(n). We show that the restrictions phi(j)\Z(n), j greate r-than-or-equal-to 1, axe weak type (1, 1) multipliers for L1 (T(n)). Moreover, the weak type (1, 1) norm of the maximal operator defined by the sequence {phi(j)}j=1infinity controls that of the maximal operato r defined by the sequence {phi(j)\Z(n)}j=1infinty. This de Leeuw type restriction theorem for maximal estimates of weak type (1, 1) answers in the affirmative a question about single multipliers posed by A. Pel czynski. Our central result, from which this restriction theorem follo ws by suitable regularization arguments, is another maximal theorem re garding convolution of a function in L1 (R(n)) with weak type (1, 1) m ultipliers.