Abstract
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 \(\ell ^\infty ({\mathbb {Z}})\).
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Communicated by Markus Haase.
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Supported by the National Natural Science Foundation of China (Grant No. 11971431), the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010006) and the first Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics).
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Ren, X., Zhang, C. Boundedness of partial difference transforms for heat semigroups generated by discrete Laplacian. Semigroup Forum 103, 622–640 (2021). https://doi.org/10.1007/s00233-021-10210-0
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DOI: https://doi.org/10.1007/s00233-021-10210-0