Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups
Let \(G\) be a locally compact Abelian group that contains a strictly decreasing sequence of open compact subgroups \((G_n)_{n=-\infty}^\infty\) such that \(\bigcup_{n=-\infty}^\infty G_n=G\), and \(\bigcap_{n=-\infty}^\infty G_n=\{0\}\). \(\Gamma\) denotes the dual group of \(G\). It is the union of the increasing sequence of its subgroups \(\Gamma_n= \{\gamma\in\Gamma: \gamma(x)=1\), \(\forall x\in G_n\}\). A complex function \(a\) is called an atom on \(G\) if \(\text{supp}(a)\subset y+G_n\), \(\|a\|_\infty\leq 1/\mu(G_n)\), \(\int_G {a(x)\,dx=0}\). The atomic Hardy space \(H^1\) consists of integrable functions \(f\) which can be represented as \(f=\sum_{j=1}^\infty \lambda_j a_j\), where each \(a_j\) is an atom and \(\sum_{j=1}^\infty|\lambda_j|<+\infty\). The norm in \(H^1\) is given by \[ \|f\|_{H^1}=\inf\sum_{j=1}^\infty |\lambda_j|, \] where the infimum is taken over all such decompositions of \(f\). The authors provide some characterization of atomic Hardy spaces on unbounded locally compact Vilenkin groups by means of a modified maximal function. The results are of independent interest, and make possible to reduce the investigation to the spaces defined by these maximal functions. The main results of the article are certain sufficient conditions on the kernel of the multiplier operator. Let us cite one of the results obtained. Theorem. Let \(\varphi\in L^\infty (\Gamma)\) and \(\sup_N\int_{G_N^c}|(\varphi-\varphi_{N+1})^\vee (y)|\,dy=O(1)\), where \(\varphi_{N+1}=\varphi{\mathbf{1}}_{\Gamma_{N+1}}\) and \(\vee\) denotes the inverse Fourier transform. Then, \(\varphi\) is a multiplier in \(H^1\), i.e., \(\varphi\in m(H^1)\). The article also contains an interesting comparison with preceding known results. However, when we have some sufficient condition for a multiplier, the question arises of how sharp it is, i.e., about some necessary conditions. It seems to be interesting and is a good question for further investigations. A bibliography reflects several results obtained in this area. The article should be interesting for specialists in harmonic analysis, operator theory and other connected areas.
scientific article; zbMATH DE number 5651214