Boundedness of partial difference transforms for heat semigroups generated by discrete Laplacian

Abstract

We analyze the convergence of the following type of series

$$\begin{aligned} T_N f(n)=\sum _{j=N_1}^{N_2} v_j\Big (e^{a_{j+1}{\varDelta }_d} f(n)-e^{a_{j}{\varDelta }_d} f(n)\Big ),\quad n\in {\mathbb {Z}}, \end{aligned}$$

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 $\ell ^\infty ({\mathbb {Z}})$.