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.


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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