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Integral means and boundary limits of Dirichlet series. (English) Zbl 1180.30002

By the Carlson theorem it is known that if a Dirichlet series \[ f(s)=\sum\limits_{n=1}^\infty a_n n^{-s}, \quad s=\sigma+it, \] converges for \(\sigma>0\) and is bounded for \(\sigma\geq\delta>0\), then, for \(\sigma>0\), \[ \lim\limits_{T\to\infty}{1\over T}\int_0^T|f(\sigma+it)|^2 dt = \sum\limits_{n=1}^\infty|a_n|^2 n^{-2\sigma}. \]
In [H. Hedenmalm, Dirichlet series and functional analysis. Laudal, Olav Arnfinn (ed.) et al., The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3-8, 2002. Berlin: Springer. 673–684 (2004, Zbl 1068.30005)], the following problem was posed: suppose that the function \(f(s)\) is bounded for \(\sigma>0\), is the above equality true for \(\sigma=0\)? The authors prove that the answer is negative. More precisely, let \(\mathcal{H}^\infty\) be the class of functions that are given by Dirichlet series in some half-plane and are bounded for \(\sigma>0\). Then they prove that there exists a function \(f\in \mathcal{H}^\infty\) such that the limit \[ \lim\limits_{T\to\infty}{1\over T}\int_0^T|f(it)|^2 dt \] does not exist, and, for given \(\epsilon>0\), there exists a singular inner function \(f\in \mathcal{H}^\infty\) such that \[ \left(\sum\limits_{n=1}^\infty |a_n|^2 \right)^{1\over 2}\leq \epsilon. \] Moreover, in the paper an embedding problem for the space \(\mathcal{H}^p\) , \(1\leq p<\infty\), introduced in [F. Bayart, Monatsh. Math. 136, No. 3, 203–236 (2002; Zbl 1076.46017)] is discussed, and versions of Fatou’s theorem are presented.

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
42B30 \(H^p\)-spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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