We analyze the convergence of the following type of series

where \(\{e^{t{\varDelta }_d} \}_{t>0}\) is the heat semigroup of the discrete Laplacian \({\varDelta }_d\), \(N=(N_1, N_2)\in {\mathbb {Z}}^2\) with \(N_1<N_2\), \(\{v_j\}_{j\in {\mathbb {Z}}}\) is a bounded real sequence and \(\{a_j\}_{j\in {\mathbb {Z}}}\) is an increasing real sequence. Our analysis will consist in the boundedness on \(\ell ^p({\mathbb {Z}}, \omega )\) of the operators \(T_N\) and its maximal operator \(\displaystyle T^*f(n)= \sup _N \left| T_N f(n)\right| \), where \(1\le p<\infty \) and \(\omega \) is a discrete Muckenhoupt weight. Moreover, we also get the alternative behavior of the maximal operator \(T^*\) on \(\ell ^\infty ({\mathbb {Z}})\).

我们分析以下类型序列的收敛性

其中\（\ {E 1 {吨{\ varDelta} _d} \} _ {T> 0} \）是离散拉普拉斯热半群\（{\ varDelta} _D \），\（N =（N_1， N_2)\in {\mathbb {Z}}^2\)与\(N_1<N_2\)，\(\{v_j\}_{j\in {\mathbb {Z}}}\)是有界实数序列和\(\{a_j\}_{j\in {\mathbb {Z}}}\)是递增的实数序列。我们的分析将包括在所述有界\（\ ELL ^ P（{\ mathbb {Z}}，\欧米加）\）的运营商的\（T_N \）和它的最大操作符\（\的DisplayStyle T 1 * F（N )= \sup _N \left| T_N f(n)\right| \)，其中\(1\le p<\infty \)和\(\omega \)是离散的 Muckenhoupt 权重。此外，我们还得到了极大运算符\(T^*\)在\(\ell ^\infty ({\mathbb {Z}})\) 上的替代行为。