Restrictions from ${\displaystyle {ℝ}^n}$ to ${\displaystyle {\mathbb{Z}}^n}$ of weak type (1,1) multipliers

Suppose that ${\displaystyle {\left\{{\varphi }_j\right\}}_{j=1}^{\infty }}$ is a sequence of weak type (1, 1) multipliers for ${\displaystyle L^1\left(R^n\right)}$ such that for each ${\displaystyle j}$, ${\displaystyle {\varphi }_j}$ is continuous at every point of ${\displaystyle Z^n}$. We show that the restrictions ${\displaystyle {\varphi }_j\mid Z^n,j\ge 1}$, are weak type (1, 1) multipliers of ${\displaystyle L^1\left(T^n\right)}$. Moreover, the weak type (1, 1) norm of the maximal operator defined by the sequence ${\displaystyle {\left\{{\varphi }_j\mid Z^n\right\}}_{j=q}^{\infty }}$ controls that of the maximal operator defined by the sequence ${\displaystyle {\left\{{\varphi }_j\mid Z^n\right\}}_{j=1}^{\infty }}$. 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. Pełczyński. Our central result, from which this restriction theorem follows by suitable regularization arguments, is another maximal theorem regarding convolution of a function in ${\displaystyle L^1\left(R^n\right)}$ with weakt type (1, 1) multipliers.