\(C^1\)-approximation and extension of subharmonic functions (Q2759307)

The authors prove criteria for the uniform approximability in \(\mathbb R^n\), \(N\geq 2\), of the gradients of \(C^1\)-subharmonic functions by the gradients of similar functions that are harmonic in neighbourhoods of a fixed compact set: Let \(f\) be a subharmonic function with continuous differential on \(\mathbb R^n\), and \(\mu = \Delta f\) its Riesz measure. Let \(X\) be a compact subset of \(\mathbb R^n\). Then the following conditions are equivalent. (i) There exists a sequence \(\{f_n\}_{n=1}^\infty\) (each \(f_n\) is a subharmonic function with continuous differential in \(\mathbb R^n\) that is harmonic in some neighbourhood of \(X\) depending on \(n\)) such that \(\nabla f_n\to\nabla f\) uniformly on \(\mathbb R^N\) as \(n\to\infty\). (ii) For each open bounded set \(D\) we have \(\mu(D)\leq C \text{osc}(\nabla f,2B) \gamma_+(D\setminus X),\) where \(B\) is a ball containing \(D\) and \(C > 0\) is independent of \(f\) and \(D\), and \(\gamma_+(E)\) denotes the capacity of a set \(E\). (iii) There exist \(k\geq 1\) and \(\varepsilon(t)\), \(\varepsilon(t)\to 0\) as \(t\to 0\), such that \(\mu(B)\leq\varepsilon(r) \gamma_+(kB\setminus X)\) for each open ball \(B\) of radius \(r\leq 1\) intersecting \(X\). (iv) There exist \(k\geq 1\) and \(\varepsilon(t)\), \(\varepsilon(t)\to 0\) as \(t\to 0\), such that NEWLINE\[NEWLINE0\leq\frac{1}{\sigma(\partial B)}\int_{\partial B}f(x) d\sigma(x)-\frac{1}{|B|}\int_Bf(x) dx\leq\varepsilon(r)r^{2-N}\gamma_+ (kB\setminus X)NEWLINE\]NEWLINE for each open ball \(B\) of radius \(r\leq 1\) intersecting \(X\), where \(\sigma\) is the surface measure on \(\partial B\) and \(|B|\) is the \(N\)-dimensional volume of \(B\). The semiadditivity properties of the capacities related to the problem are proved. It is shown that these properties give sufficient geometric conditions for the approximability of \textit{classes} of functions. An estimate of the flux of the gradient of a subharmonic function in terms of the capacity of its `sources' and the theorem on the possibility of a \(C^1\)-extension of a subharmonic function in a ball to a subharmonic function on the whole of \(\mathbb R^n\) are established: Let \(f\) be a subharmonic function in an open ball \(B\) that is of class \(C^1\) on \(\overline B\). Then there exists a function \(F\), subharmonic and of class \(C^1\) in \(\mathbb R^N\) such that \(F=f\) on \(\overline B\) and \(\|\nabla F\|_{\mathbb R^N}\leq C\|\nabla f\|_B\), where \(C\) is a positive constant depending only on \(N\). The \(T(1)\)-theorem due to G.~David and J.~L.~Journé is discussed.