Multipliers on weighted Hardy spaces over certain totally disconnected groups (Q1113414)

Let G be a totally disconnected, locally compact abelian group. Certain weighted function spaces over G are introduced: \(L^ 1_{\alpha}(G)\) and \(H^ 1_{\alpha}(G)\subset L^ 1_{\alpha}(G)\). Let the symbols \(\wedge\) and \(\vee\) denote Fourier and inverse Fourier transform, respectively. A function \(m\in L^{\infty}(G)\) is called an (X,Y)- multiplier if there exists a constant C such that \(\| m\phi^{\wedge})^{\vee}\|_ Y\leq C\| \phi \|_ X\), for all test functions \(\phi\). The results of this paper consist in a description of certain \((H^ 1_{\alpha},L^ 1_{\alpha})\)-multipliers as well as certain \((H^ 1_{\alpha},H^ 1_{\alpha})\)-multipliers. The details are lengthy.