×

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
PDF BibTeX XML Cite
Full Text: EuDML EMIS