Discrete Hardy spaces related to powers of the Poisson kernel (Q2847667)

The paper deals with the study of Hardy spaces in the discrete setting that corresponds to rooted regular trees \(T\) of fixed degree \(q+1\geq 3\). Denoting by \(\partial T\) the boundary of the tree, for a given \(f\in L^1(\partial T)\) with respect to the harmonic measure \(\omega\) on \(\partial T\) , \(\alpha \in [1/2,+\infty)\) and \(x\in T\), the action of the power \(\alpha\) of the Poisson kernel is defined as follows NEWLINE\[NEWLINE(P_{\alpha} f)(x)=\int_{\partial T} (q^{2N(x,\xi)-|x|})^{\alpha} \;f(\xi) \;d\omega(\xi),NEWLINE\]NEWLINE where \(N(x,\xi)\) is the number of edges shared by the geodesic from the root to \(x\), having \(|x|\) edges, and \(\xi\). The normalized \(\alpha\)-Poisson extension of \(f\) to \(T\) is defined by NEWLINE\[NEWLINE({\mathcal P}_{\alpha} f)(x)= \frac{(P_{\alpha} f)(x)}{(P_{\alpha} 1)(x)}NEWLINE\]NEWLINE for \(x\in T\), and the corresponding radial maximal function is NEWLINE\[NEWLINE ({\mathcal P}_{\alpha}^* f)(\xi)=\sup_{n\in \mathbb{N}} |({\mathcal P}_{\alpha} f)(\xi_n)|\quad \text{for all \(\xi \in \partial T\),} NEWLINE\]NEWLINE where \(\xi_n\) is the vertex along \(\xi\) which is \(n\) edges away from the root.NEWLINENEWLINEFor \(\alpha \in [1/2,+\infty)\), \(H^1_{\alpha}(\partial T)\) denotes the \(\alpha-\)Hardy space which is defined as the subspace of \(L^1(\partial T)\) such that the radial maximal function \({\mathcal P}_{\alpha}^* f\) belongs to \(L^1(\partial T)\) and is endowed with the norm NEWLINE\[NEWLINE\|f\|_{H^1_{\alpha}(\partial T)}=\|{\mathcal P}_{\alpha}^* f\|_{L^1(\partial T)}.NEWLINE\]NEWLINENEWLINENEWLINEThe paper shows that, for \(\alpha > 1/2\), the spaces coincides with the ordinary atomic Hardy space on \(\partial T\) of Coifman and Weiss and they strictly contain \(H^1_{1/2}(\partial T)\). It is also proved that the Orlicz space \(L \log \log L (\partial T)\) characterizes the positive and increasing functions in \(H^1_{1/2}(\partial T)\), but there is no characterization for general positive functions.