A theorem of Hardy-Littlewood and removability for polyharmonic functions satisfying Hölder's condition (Q1894034)

The author proves the following polyharmonic analogue of a theorem first proved by Hardy and Littlewood for holomorphic functions. Let \(u\) be a polyharmonic function on the unit ball \(B\) of \(\mathbb{R}^n\). If \(u\) is in the Hölder class \(\Lambda_\alpha (B)\), where \(0<\alpha\leq 1\), then \(|\text{grad } u(x) |\leq M(1- |x|^2 )^{\alpha- 1}\) for each \(x\in B\) and some constant \(M\). The above result is used to prove continuation theorems for polyharmonic functions. Denoting \(\beta\)-dimensional Hausdorff measure by \(H_\beta\), we now state one such theorem: Let \(K\) be a compact subset of an open set \(G\) in \(\mathbb{R}^n\), and let \(u\) be polyharmonic of order \(m\) on \(G\setminus K\). If \(u\in \Lambda_\alpha (G)\), where \(2m-n< \alpha< 2m\), and \(H_{n+ \alpha- 2m} (K) =0\), then \(u\) can be extended to become polyharmonic of order \(m\) on \(G\). This continuation theorem generalizes a result of \textit{D. C. Ullrich} [Mich. Math. J. 38, 467-473 (1991; Zbl 0751.31001)]\ on harmonic continuation.