Hardy spaces for the strip
For computational convenience, the authors study Hardy spaces on the strip \(\mathfrak{S} = \{ z \in \mathbb{C} \,: | \text{Im}\, z | < 1\}\). They consider three candidates to be genuine Hardy spaces on such strip. Firstly, since \(\phi(z) = \tanh \,\frac{\pi}{4}\) is a conformal maps from \(\mathfrak{S}\) onto the unit disc \(\mathbb{D}\), it is natural to define \(H^p(\mathfrak{S}) = \{ f \in \text{Hol}(\mathbb{D}) \,:\, f \circ \phi^{-1} \in H^p({\mathbb{D}})\}\), \(0< p < \infty\), and \(\| f| _{H^p(\mathfrak{S})} = \| f \circ \phi^{-1}\| _{H^p({\mathbb{D}})}\). A second class of \(H^p\)-spaces are the translation invariant spaces defined as \[ H^p_{\tau} (\mathfrak{S}) = \left\{f \in \text{Hol} (\mathfrak{S})\,: \, \| f\| ^p_{H^p_{\tau}(\mathfrak{S})} = \sup_{0 \leq y < 1} \int_{-\infty}^\infty \frac{| f(-iy + t)| ^p + | f(iy + t)| ^p}{2}\,dt < \infty \right\}, \] which sometimes are studied in the context of Paley-Wiener theory. A third class, defined as \[ H^p_{\mathcal{W}}(\mathfrak{S}) = \left\{f \in \text{Hol} (\mathfrak{S})\,: \, \| f\| ^p_{H^p_{\mathcal{W}}(\mathfrak{S})} = \sup_{0 < y < 1} \int_{-\infty}^\infty \frac{| f(-iy + t)| ^p + | f(iy + t)| ^p}{4(\cosh \frac{\pi}{2}t + \cos\frac{\pi}{2}y ) }\,dt < \infty \right\}, \] is related to the Poisson measure for the origin. The main results of the paper state that for \(0 < p < \infty\), the spaces \(H^p_{\mathcal{W}}(\mathfrak{S})\) and \(H^p_{\tau} (\mathfrak{S})\) are isometrically isomorphic (Theorem 2.1), that the spaces \(H^p_{\mathcal{W}}(\mathfrak{S})\) and \(H^p(\mathfrak{S})\) are identical with identical norms (Theorem 2.2), that functions in \(H^p_{\tau} (\mathfrak{S})\) satisfy a \(L^p\)-form of the Hadamard three lines theorem (Theorem 2.3), and that algebraic polynomials are dense in \(H^p (\mathfrak{S})\) (Theorem 2.4). It is said that an orthogonal basis of polynomials for \(H^2(\mathfrak{S})\) is presented. However, this interesting matter is just sketched in the last section. Reviewer's remark: Most techniques used are classical and standard, and I should also point out that I have not been able to find out in the literature the reference no. 16. In fact, to best of my knowledge, the well known \textit{Trigonometric Series} Zygmund's book has only two volumes [cf. \textit{A. Zygmund}, Trigonometric series. Volumes I and II combined. With a foreword by Robert Fefferman. 3rd ed. Cambridge Mathematical Library. (Cambridge): Cambridge University Press. (2002; Zbl 1084.42003)].
scientific article; zbMATH DE number 5167521
scientific article