The decomposition of Herz spaces on local fields and its applications (Q1910058)

Let \(\mathbb{K}\) be a local field with modular function \(x \to |x |\), where \(|x |= q^n\) for some \(q \in \mathbb{N}\) and \(n \in \mathbb{Z}\). Let \(\mu\) be the Haar measure on \((K, +)\), chosen so that for each \(B^n : = \{x \in \mathbb{K} : |x |< q^{- n}\}\) we have \(\mu (B^n) = q^{- n}\). An \((\alpha, p)\)-unit (restricted \((\alpha, p)\)-unit) is a measurable function \(b\) on \(K\) such that (i) \(\text{supp} b \subset B^n\) for some \(n \in \mathbb{Z}\) \((n \in \mathbb{Z} \backslash \mathbb{N})\), and (ii) \(|b |_p \leq q^{n \alpha}\). The authors present a decomposition theorem for the elements of certain homogeneous Herz space \(\dot K (\alpha, p, \ell)\) and nonhomogeneous Herz spaces \(K (\alpha, p, \ell)\) on \(\mathbb{K}\), into \((\alpha, p)\)-units and restricted \((\alpha, p)\)-units, respectively. Using this decomposition result, the following theorem is proved. Let \(1 \leq p_0 \leq p < \infty\), let \(0 < \alpha < (1/p_0) (1 - 1/p)\) and \(0 < \ell \leq \infty\). If \(T : L^p (\mathbb{K}) \to L^p (\mathbb{K})\) is a sublinear operator so that \(|T f(x) |^{p_0} \leq C \int_\mathbb{K} |f(y) |^{p_0}/ |x - y |dy\) for all \(f \in L^p (\mathbb{K})\) with compact support, then \(T\) maps \(\dot K (\alpha, p, \ell)\) into \(\dot K (\alpha, p, \ell)\) and \(K (\alpha, p, \ell)\) into \(K (\alpha, p, \ell)\). -- Moreover, it is shown that the range of \(\alpha\) cannot be extended.