In this case the arguments are similar to those in the centered case, but the arguments are more complex than those in the centered case. We want to show that
$$ \bigl\Vert D_{d}\widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigr\Vert _{\ell^{1}(\mathbb{Z}^{d})} \lesssim_{\alpha,m,d}\prod _{i=1}^{m} \Vert f_{i} \Vert _{\ell^{1}(\mathbb{Z}^{d})}. $$
(3.8)
For each
\(n'\in\mathbb{Z}^{d-1}\), let
$$\begin{aligned}& Y_{n'}=\bigl\{ n_{d}\in\mathbb{Z}: \widetilde{ \mathbf{M}}_{\alpha}(\vec {f}) \bigl(n',n_{d}+1 \bigr)=\widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d} \bigr)\bigr\} , \\& Y_{n'}^{+}=\bigl\{ n_{d}\in\mathbb{Z}: \widetilde{\mathbf{M}}_{\alpha}(\vec {f}) \bigl(n',n_{d}+1 \bigr)>\widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d} \bigr)\bigr\} , \\& Y_{n'}^{-}=\bigl\{ n_{d}\in\mathbb{Z}: \widetilde{\mathbf{M}}_{\alpha}(\vec {f}) \bigl(n',n_{d}+1 \bigr)< \widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d} \bigr)\bigr\} . \end{aligned}$$
Fix
\(n'\in\mathbb{Z}^{d-1}\). Since all
\(f_{j}\in\ell^{1} (\mathbb{Z}^{d})\), then for any
\(n_{d}\in Y_{n'}^{+}\), there exist
\(r(n',n_{d}+1)>0\) and
\(\vec{l}\in \mathbb{Z}^{d}\) such that
\(\widetilde{\mathbf{M}}_{\alpha}(\vec{f})(n',n_{d}+1)=A_{r(n',n_{d}+1)}(\vec{f})(\vec{l})\) and
\(\Vert(n',n_{d}+1)-\vec{l} \Vert_{1}< r(n',n_{d}+1)\). By the arguments similar to those used in deriving (
3.5), we obtain
$$ \begin{aligned}[b] &\widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d}+1\bigr)-\widetilde {\mathbf{M}}_{\alpha}( \vec{f}) \bigl(n',n_{d}\bigr) \\ &\quad\leq A_{r(n',n_{d}+1)}(\vec{f}) (\vec{l})-A_{r(n',n_{d}+1)+1}(\vec {f}) ( \vec{l}-\vec{e}_{d}) \\ &\quad\leq\Phi\bigl(\bigl[r\bigl(n',n_{d}+1\bigr)\bigr] \bigr)\prod_{j=1}^{m}\sum _{\vec{k}\in\Gamma _{r(n',n_{d}+1)}(\vec{l})}f_{j}(\vec{k}) \\ &\quad\leq\Biggl(\prod_{i=1}^{m-1} \Vert f_{i} \Vert _{\ell ^{1}(\mathbb{Z}^{d})}\Biggr)\sum_{\vec{k}\in\Gamma _{2r(n',n_{d}+1)}(n',n_{d}+1)} \Phi_{m-\frac{\alpha }{d}}\bigl(\bigl[r\bigl(n',n_{d}+1\bigr) \bigr]\bigr)f_{m}(\vec{k}). \end{aligned} $$
(3.9)
Note that
\(8[r]\geq[2r]\) for
\(r\geq2\) and
\(\Phi(r)\leq1\) for all
\(r\in\mathbb{N}\). By Lemma
2.1, one can get that
$$\begin{aligned} &\sum_{\vec{k}\in\Gamma_{2r(n',n_{d}+1)}(n',n_{d}+1)} \Phi_{m-\frac {\alpha}{d}}\bigl(\bigl[r\bigl(n',n_{d}+1\bigr) \bigr]\bigr)f_{m}(\vec{k}) \\ &\quad\lesssim_{\alpha,m,d}\sum_{\vec{k}\in\Gamma _{2r(n',n_{d}+1)}(n',n_{d}+1)} \Phi_{m-\frac{\alpha }{d}}\bigl(8\bigl[r\bigl(n',n_{d}+1\bigr) \bigr]\bigr)f_{m}(\vec{k}) \\ &\quad\lesssim_{\alpha,m,d}\sum_{\vec{k}\in\Gamma _{2r(n',n_{d}+1)}(n',n_{d}+1)}f_{m}( \vec{k})\chi_{\{ \Vert \vec {k}-(n',n_{d}+1) \Vert _{1}< 2r(n',n_{d}+1)< 4\}}\bigl(n',n_{d}\bigr) \\ &\hphantom{\quad\lesssim_{\alpha,m,d}}{}+\sum_{\vec{k}\in\Gamma_{2r(n',n_{d}+1)}(n',n_{d}+1)}\Phi _{m-\frac{\alpha}{d}}\bigl( \bigl[2r\bigl(n',n_{d}+1\bigr)\bigr]\bigr)f_{m}( \vec{k})\chi_{\{ r(n',n_{d}+1)\geq2\}}\bigl(n',n_{d}\bigr) \\ &\quad\lesssim_{\alpha,m,d}\sum_{\vec{k}\in\mathbb{Z}^{d}}f_{m}( \vec {k})\chi_{\{ \Vert \vec{k}-(n',n_{d}+1) \Vert _{1}< 4\} }\bigl(n',n_{d}\bigr) \\ &\hphantom{\quad\lesssim_{\alpha,m,d}}{}+\sum_{\vec{k}\in\mathbb{Z}^{d}}f_{m}(\vec{k}) \Phi_{m-\frac {\alpha}{d}}\bigl(\bigl[2r\bigl(n',n_{d}+1\bigr) \bigr]\bigr)\chi_{\{ \Vert \vec{k}-(n',n_{d}+1) \Vert _{1}\leq[2r(n',n_{d}+1)]\}}\bigl(n',n_{d}\bigr). \end{aligned}$$
(3.10)
By the arguments similar to those used to derive (
3.7), we get
$$ \begin{aligned}[b] &\sup_{\vec{k}\in\mathbb{Z}^{d}}\sum_{n'\in\mathbb{Z}^{d-1}}\sum _{n_{d}\in X_{n'}^{+}}\Phi_{m-\frac{\alpha}{d}}\bigl(\bigl[2r \bigl(n',n_{d}+1\bigr)\bigr]\bigr)\chi _{\{ \Vert \vec{k}-(n',n_{d}+1) \Vert _{1}\leq [2r(n',n_{d}+1)]\}} \bigl(n',n_{d}\bigr)\\ &\quad \lesssim_{\alpha,m,d}1. \end{aligned} $$
(3.11)
It follows from (
3.9)–(
3.11) that
$$ \begin{aligned}[b] &\sum_{n'\in\mathbb{Z}^{d-1}}\sum _{n_{d}\in Y_{n'}^{+}}\bigl(\widetilde {\mathbf{M}}_{\alpha}( \vec{f}) \bigl(n',n_{d}+1\bigr)-\widetilde{\mathbf {M}}_{\alpha}(\vec{f}) \bigl(n',n_{d}\bigr)\bigr) \\ &\quad\lesssim_{\alpha,m,d}\Biggl(\prod_{i=1}^{m-1} \Vert f_{i} \Vert _{\ell^{1}(\mathbb{Z}^{d})}\Biggr)\\ &\hphantom{\quad\lesssim_{\alpha,m,d}} {}\times \biggl(\sum _{n'\in\mathbb {Z}^{d-1}}\sum_{n_{d}\in Y_{n'}^{+}}\sum _{\vec{k}\in\Gamma _{2r(n',n_{d}+1)}(n',n_{d}+1)}f_{m}(\vec{k})\chi_{\{ \Vert \vec {k}-(n',n_{d}+1) \Vert _{1}< 4\}} \bigl(n',n_{d}\bigr) \\ &\hphantom{\quad\lesssim_{\alpha,m,d}}{}+\sum_{n'\in\mathbb{Z}^{d-1}}\sum _{n_{d}\in Y_{n'}^{+}}\sum_{\vec{k}\in\mathbb{Z}^{d}}f_{m}( \vec{k})\Phi_{m-\frac{\alpha }{d}}\bigl(\bigl[2r\bigl(n',n_{d}+1 \bigr)\bigr]\bigr)\\ &\hphantom{\quad\lesssim_{\alpha,m,d}} {}\times\chi_{\{ \Vert \vec{k}-(n',n_{d}+1) \Vert _{1}\leq[2r(n',n_{d}+1)]\}}\bigl(n',n_{d}\bigr) \biggr) \\ &\quad\lesssim_{\alpha,m,d}\Biggl(\prod_{i=1}^{m} \Vert f_{i} \Vert _{\ell^{1}(\mathbb{Z}^{d})}\Biggr) \biggl(\sup _{\vec{k}\in\mathbb{Z}^{d}}\sum_{n'\in\mathbb{Z}^{d-1}}\sum _{n_{d}\in Y_{n'}^{+}}\chi_{\{ \Vert \vec{k}-(n',n_{d}+1) \Vert _{1}< 4\}}\bigl(n',n_{d} \bigr) \\ &\hphantom{\quad\lesssim_{\alpha,m,d}}{}+\sup_{\vec{k}\in\mathbb{Z}^{d}}\sum_{n'\in\mathbb {Z}^{d-1}} \sum_{n_{d}\in Y_{n'}^{+}}\Phi_{m-\frac{\alpha }{d}}\bigl(\bigl[2r \bigl(n',n_{d}+1\bigr)\bigr]\bigr)\\ &\hphantom{\quad\lesssim_{\alpha,m,d}}{}\times\chi_{\{ \Vert \vec{k}-(n',n_{d}+1) \Vert _{1}\leq[2r(n',n_{d}+1)]\}} \bigl(n',n_{d}\bigr)\biggr) \\ &\quad\lesssim_{\alpha,m,d}\prod_{i=1}^{m} \Vert f_{i} \Vert _{\ell^{1}(\mathbb{Z}^{d})}. \end{aligned} $$
(3.12)
Similarly, we can obtain
$$ \sum_{n'\in\mathbb{Z}^{d-1}}\sum_{n_{d}\in Y_{n'}^{-}} \bigl(\widetilde {\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d} \bigr)-\widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d}+1 \bigr)\bigr)\lesssim_{\alpha,m,d}\prod_{i=1}^{m} \Vert f_{i} \Vert _{\ell^{1}(\mathbb{Z}^{d})}. $$
(3.13)
It follows from (
3.12) and (
3.13) that
$$\begin{aligned}[b] \bigl\Vert D_{d}\widetilde{ \mathbf{M}}_{\alpha}(\vec{f}) \bigr\Vert _{\ell^{1}(\mathbb{Z}^{d})}&=\sum _{n'\in\mathbb{Z}^{d-1}}\sum_{n_{d}\in Y_{n'}^{+}}\bigl(\widetilde{ \mathbf{M}}_{\alpha}(\vec {f}) \bigl(n',n_{d}+1 \bigr)-\widetilde{\mathbf{M}}_{\alpha}(\vec{f}) \bigl(n',n_{d} \bigr)\bigr) \\ &\quad{}+\sum_{n'\in\mathbb{Z}^{d-1}}\sum _{n_{d}\in Y_{n'}^{-}}\bigl(\widetilde{\mathbf{M}}_{\alpha}(\vec {f}) \bigl(n',n_{d}\bigr)-\widetilde{\mathbf{M}}_{\alpha}( \vec{f}) \bigl(n',n_{d}+1\bigr)\bigr) \\ &\lesssim_{\alpha,m,d}\prod_{i=1}^{m} \Vert f_{i} \Vert _{\ell^{1}(\mathbb{Z}^{d})}. \end{aligned} $$
This proves (
3.8) and completes the proof of the boundedness part.