## 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.

### MSC:

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