On removability of sets for holomorphic and harmonic functions (Q1820273)

For a function \(f\) defined on an open set \(W\) of \({\mathbb{C}}\), \(S(f)\) denotes the set of all points at which \(f\) fails to admit a complex derivative. Also, for a nondecreasing function \(h\) on \([0,\infty)\) with \(h(0)=0\), \(h(r)>0\) for \(r>0\) and \(h(2r)\leq\text{constant } h(r)\), \(r>0\), let \(\Lambda_ h\) denote the Hausdorff measure associated with \(h\). For \(h(r)=r^{\alpha}\), \(\alpha >0\), denote \(\Lambda_ h\) by \(\Lambda_{\alpha}\). Let \(1\leq p\leq \infty\) and \(1/p+1/q=1\). If f is locally integrable on \(W\), define \[ F(z)=\sup_{B}r^{-1-2/p} h(r)^{1/q}\inf_{g}\int_{B}| f(w)-g(w)| d\Lambda_ 2(w) \] where the supremum is taken over all open discs \(B\) with radius r such that \(z\in B\subset W\), and the infimum is taken over all holomorphic functions \(g\) in \(B\). In the paper, the author proves the following Theorem. Supose \(F\in L^ p(W).\) (i) If \(p<\infty\), \(\lim_{r\to 0}r^{-2}h(r)=\infty\) and \(\Lambda_ h(S(f))<\infty\), then \(f\) can be corrected on a set of measure zero to be holomorphic in \(W\). (ii) If \(p=1\) and \(\Lambda_ 2(S(f))=0\) or if \(p>1\) and \(\Lambda_ h(S(f))=0\), then the same conclusion holds. Analogous results are also obtained for subharmonic functions in \({\mathbb{R}}^ n\).