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Estimate for gradient, BMO and Lindelöf theorem. (English) Zbl 0944.31004

Let \(u_{\beta}\) denote the Bloch norm of a function \(u\in C^1(D)\), where \(D\) is a proper subdomain of \(\mathbb{R}^n\), and let \(u_*\) be its BMO norm.
It is proved that if \(u\) is a harmonic function on \(D\) then \(u_{\beta}\leq (n+1) u_*\), and if \(u\in C^1(D)\), then \(u_*\leq b_n u_{\beta}\), where \(b_n=1+\frac 12+\cdots + \frac 1n\).
Some applications of these estimates are given.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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