## Removable singularities for $$H^p$$ spaces of analytic functions, $$0<p<1$$.(English)Zbl 1013.30020

For $$\Omega$$ a domain in $$\mathbb{C}$$, the Hardy space $$H^p (\Omega)$$ is the space of holomorphic functions $$f :\Omega\to\mathbb{C}$$ such that there exists a harmonic function $$u : \Omega\to\mathbb{R}^+$$ with $$|f|^p\leq u$$. A compact set $$K\subset\mathbb{C}$$ is said to be $$H^p$$-removable if there exists a bounded neighbourhood $$\Omega$$ of $$K$$ such that any function $$f\in H^p(\Omega\setminus K)$$ extends to a holomorphic function on $$\Omega$$ (then this holds for any neighbourhood of $$K$$). This notion extends to non compact sets, but not in a unique way: one has to distinguish between weak removability and strong removability.
The author investigates $$H^p$$-removability for $$0<p<1$$. The case $$1\leq p\leq\infty$$ has been studied in earlier papers; most results are classical for $$p=\infty$$ (denoting by $$H^p(\Omega)$$ the space of bounded holomorphic functions on $$\Omega$$). He shows that, for any $$p$$ with $$0<p<1$$ , there is a compact set $$K$$ which is $$H^p$$-removable, with Hausdorff dimension $$\dim(K) >0$$. Such examples were not known before. For $$p\leq 0.65$$ , he gets $$\dim(K) > {1\over 2}p$$. It is known that $$H^p$$-removability implies $$\dim(K)\leq p$$.
The examples he gives are just affine Cantor sets in $$\mathbb{R}$$. For $$0<\alpha<{1\over 2}$$ , set $$C_\alpha^1 = [0,\alpha]\cup [1-\alpha ,1]$$, and define $$F_\alpha : C_\alpha^1 \to [0,1]$$ to be the map which induces increasing affine bijections $$[0,\alpha]\to [0,1]$$ and $$[1-\alpha ,1] \to [0,1]$$. The affine Cantor set $$C_\alpha$$ is the set of points $$x\in C_\alpha^1$$ whose orbit under $$F_\alpha$$ is defined for all $$n$$ and stays in $$C_\alpha^1$$. The Hausdorff dimension of $$C_\alpha^1$$ is $$\log 2 /\log({1\over\alpha})$$. The author shows that $$C_\alpha$$ is $$H^p$$-removable for some $$p(\alpha)$$ (and naturally for $$p\geq p(\alpha)$$). He gives algorithms to compute $$p(\alpha)$$, and numerical tables. He gives examples which show that $$H^p$$-removability does not behave like usual “smallness” properties. For instance, one can find $$K_1$$ and $$K_2$$ which are $$H^{1\over 2}$$-removable, but such that $$K_1\cup K_2$$ is not $$H^p$$-removable for any $$p <1$$. He finally states some conjectures.

### MSC:

 30D55 $$H^p$$-classes (MSC2000) 30B40 Analytic continuation of functions of one complex variable 30C85 Capacity and harmonic measure in the complex plane 46E10 Topological linear spaces of continuous, differentiable or analytic functions
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