On Hardy classes of analytic functions with a variable exponent. (English) Zbl 1126.47031

Let \(U= \{z\in \mathbb C:|z|< 1\}\) be the unit ball with boundary \(\gamma\), and \(p= p(t)\), \(t\in\gamma\), be a bounded, positive, measurable function. An analytic function \(\varphi\) on \(U\) is said to belong to the Hardy class \(H^{p(.)}\) if \[ \sup_{r< 1}\;\int^{2\pi}_0 |\varphi(re^{i\theta})|^{p(\theta)}\,d\theta= C< \infty, \] where \(p(\theta)= p(e^{i\theta})\), \(\theta\in [0, 2\pi)\).
Similarly, if \(u= u(re^{i\theta})\) is harmonic on \(U\) and \[ \sup_{r< 1}\;\int^{2\pi}_0 |u(re^{i\theta})|^{p(\theta)}\,d\theta= C<\infty, \] then \(u\) is said to be in the class \(h^{p(.)}\).
Without giving proofs, the authors state six theorems related to the classes \(H^{p(.)}\) and \(h^{p(.)}\), which are extensions of some well-known results for the case when \(p(\theta)= p\) is a positive constant.


47B38 Linear operators on function spaces (general)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
45P05 Integral operators