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\(\mathcal{H}_p \)-theory of general Dirichlet series. (English) Zbl 1429.43004

The authors introduce a Dirichlet group \((G,\beta)\) as a compact abelian group \(G\) together with a continuous homomorphism \(\beta: \mathbb{R} \to G\) having dense range. If \(\lambda=(\lambda_n)_n\) is a frequency (a strictly increasing sequence of non-negative real numbers) then \(G\) is said to be a \(\lambda\)-Dirichlet group if \(\lambda\subset {\hat\beta}{\hat G}\). If, in addition, \(1\le p\le\infty\), then \(H_p^\lambda(G)\) denotes the subspace of all \(f\) in \(L_p(G)\) such that \(\hat f(\gamma)\not= 0\) implies \(\gamma=h_{\lambda_n}\) for some \(n\) in \(\mathbb{N}\), where \(h_{\lambda_n}\) is the continuous extension of \(e^{-i\lambda_n}\) to \(G\). Finally, given a frequency \(\lambda\), the authors use \(H_p(\lambda)\) to denote all \(\lambda\)-Dirichlet series \(\sum_n a_n e^{-\lambda_ns}\) with there is a \(\lambda\)-Dirichlet group, \((G,\beta)\), and \(f\) in \(\mathcal{H}_p^\lambda(G)\) such that \(a_n={\hat f}(h_{\lambda_n})\) for all \(n\). It is shown that this definition is independent of the choice of the \(\lambda\)-Dirichlet group \((G,\beta)\). Examples are provided of when \(\mathcal{H}_p (\lambda)\) is isometrically isomorphic to an \(H_p\) space on products of the torus. Letting \(\mathcal{D}(\lambda)\) denote the space of all \(\lambda\)-Dirichlet series, \(\sigma_c(D)=\inf\{\sigma\in \mathbb{R}: D \text{ converges on }\{z: \mathrm{Re}(z) >\sigma\}\}\), \(\sigma_a(D)=\inf\{\sigma\in \mathbb{R}: D \text{ converges absolutely on }\{z: \mathrm{Re}(z)>\sigma\}\}\) and \(\sigma_u(D)=\inf\{\sigma \in \mathbb{R}: D \text{ converges uniformly on } \{z: \mathrm{Re}(z)>\sigma\}\}\), the authors study the relationship between \(L(\lambda)=\sup_{D\in \mathcal{D}(\lambda)} \sigma_a(D)-\sigma_c(D)\) and \(S(\lambda)=\sup_{D\in \mathcal{D}(\lambda)}\sigma_a(D) -\sigma_u(D)\) and in particular give conditions under which \(S(\lambda)=\frac{L(\lambda)}{2}\).
In the final section of the paper, the authors show: 1) \(\mathcal{H}_p( \lambda)\) is stable under ‘translations’; 2) \(\mathcal{H}_\infty(\lambda)\) is equal to the space of all \(\lambda\)-Dirichlet series which converge and define a bounded function on the half-plane \(\{z: \mathrm{Re }(z)>0\}\); 3) for \(1<p<\infty\) \(\mathcal{H}_p(\lambda)\) has a Schauder basis; 4) under certain conditions on \(\lambda\), a Montel’s theorem holds in \(\mathcal{H}_p(\lambda)\); 5) a Brother’s Riesz theorem holds for \(\mathcal{H}_1^\lambda(G)\).

MSC:

43A17 Analysis on ordered groups, \(H^p\)-theory
30H10 Hardy spaces
30B50 Dirichlet series, exponential series and other series in one complex variable
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References:

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