Illinois Journal of Mathematics

Dominating measures for spaces of analytic functions

Daniel H. Luecking

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The mixed norm space $H(p,q,a)$ is the collection of functions $f$ analytic in the unit disk with finite norm $$||f||_{p,q,\alpha}=\left[\int_{0}^{1}(1-r)^{\alpha q-1}\left(\int_{0}^{2\pi}|f\left(re^{i\theta}\right)|^{p} d\theta\right)^{q/p}dr\right]^{1/q}.$$ Sufficient conditions on a family of measures $\{\mu_{r}:0< r <1\}$ on $U$ and a measure $\nu$ on $[0,1]$ are given to obtain an inequality $$||f||_{p,q,\alpha}^{q}\leq C\int{\left(\int{|f|^{p}d\mu_{r}}\right)^{q/p}\,d\nu(r)},\quad f\in H(p,q,\alpha)$$ with $C$ independent of $f$. Similar results are obtained for spaces of ``slow mean growth'' $(q=\infty)$ and the Hardy spaces ($q=\infty$, $\alpha=0$). In the case of the Bergman spaces $(p=q)$ these conditions are an improvement over those obtained in [5] and [6].

Article information

Illinois J. Math., Volume 32, Issue 1 (1988), 23-39.

First available in Project Euclid: 19 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46J15: Banach algebras of differentiable or analytic functions, Hp-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
Secondary: 30D55 30H05: Bounded analytic functions 46E15: Banach spaces of continuous, differentiable or analytic functions


Luecking, Daniel H. Dominating measures for spaces of analytic functions. Illinois J. Math. 32 (1988), no. 1, 23--39. doi:10.1215/ijm/1255989226.

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