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John functions, quadratic integral forms and \(o\)-minimal structures. (English) Zbl 1040.31004

In this paper \(K(\Omega)\)-functions are studied via quadratic integral forms and \(o\)-minimal structures. The class \(K(\Omega)\) [F. John, Russ. Math. Surv. 29, 170–175 (1974; Zbl 0303.26011)] contains all \(C^1\) functions \(f:\Omega\to \mathbb R\) which have bounded expansion \[ \| f\|_{K(\Omega)}=\sup_{x\in\Omega}\text{ de}_{\Omega}(x) | \nabla f(x)| <+\infty. \] Here \(\Omega\) is a proper subdomain of \(\mathbb R^n, n\geq 2\), \(\partial\Omega\) is the boundary of \(\Omega\), \(\text{de}_{\Omega}(x)\) is the Euclidean distance of the point \(x\) to \(\partial\Omega\). In Section 1 some essential properties of John functions are discussed: bounded mean oscillation, global Lipschitz continuity and asymptotic behavior. In Section 2 the construction of John functions is considered. In Section 3 the results of Section 2 are applied to the \(K_G(\Omega)\)-characteristic of \(K(\Omega)\). It is assumed that \(f\in K_G(\Omega)\) provided \[ \sup_{y\in\Omega}\Bigl( \int_{\Omega}| \nabla f(x)| ^2g_{\Omega}(x,y)\,dx \Bigr)^{1/2}<+\infty, \] here \(g_{\Omega}(x,y)\) is the Green function for the Laplacian. In the last section the Harnack and Poincaré metric versions of the John functions are given.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
46E15 Banach spaces of continuous, differentiable or analytic functions
26B15 Integration of real functions of several variables: length, area, volume

Citations:

Zbl 0303.26011
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