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Subharmonic behaviour of smooth functions. (English) Zbl 0944.31003

The author considers the subharmonic behaviour of smooth functions defined in a proper subdomain \(G\) of \(\mathbb{R}^n\). The main result is as follows:
Theorem. Let \(f\in C^2(G)\). If there are nonnegative constants \(K\) and \(K_0\) such that \[ \|\Delta f(x)\|\leq {{K}\over{r}} \sup\|\nabla f\|+ {{K_0}\over{r^2}} \sup\|f\|,\qquad x\in G, \] where the supremum is taken over the Euclidean balls \(B_r (x)= \{y:\|y-x\|< r\}\subset G\), then \( \|f\|^p ,0< p < \infty \) has subharmonic behaviour, i.e. there exists a constant \(C\), \(1\leq C\), such that \[ \|f(x)\|^p \leq {{C}\over{r^n}} \int_{B_r(x)}\|f(y)\|^p dV(y),\text{ for all } x \in G, \] If in addition \(K_0=0\), then \(\|\nabla f\|^p\) has subharmonic behaviour.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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