# zbMATH — the first resource for mathematics

Entire functions of exponential type and model subspaces in $$H^ p$$. (Russian. English summary) Zbl 0788.30024
The author is concerned with norm inequalities in coinvariant subspaces of $$H^ p$$ of the upper half-plane $$(1 \leq p \leq \infty)$$. For $$\theta$$ an inner function for the upper half-plane, the corresponding coinvariant subspace of $$H^ p$$ is denoted by $$K^ p_ \theta$$ and defined to be $$H^ p \cap \theta \overline{H^ p}$$. In the special case where $$\theta(z)=e^{iaz}$$ $$(a>0)$$, the space $$K^ p_ \theta$$ is contained in the space of entire functions of exponential type at most $$a$$ that belong to $$L^ p$$ on the real axis. The author shows that several classical inequalities pertaining to the latter spaces apply also to $$K^ p_ \theta$$ provided $$\theta'$$ is in $$H^ \infty$$ (and only in that case). Sample result: If $$1<p \leq \infty$$, ten $$\| f' \|_ p \leq \text{const} \| f \|_ p$$ for all $$f$$ in $$K^ p_ \theta$$ if and only if $$\theta'$$ is in $$H^ \infty$$. This corresponds, for the case $$\theta(z)=e^{iaz}$$, to a well-known inequality of S. Bernstein.

##### MSC:
 30D55 $$H^p$$-classes (MSC2000) 46E15 Banach spaces of continuous, differentiable or analytic functions 47A99 General theory of linear operators 47B38 Linear operators on function spaces (general)
Full Text: