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Integrability of Green potentials in fractal domains. (English) Zbl 0860.31002

Let \(Gf(x)\) be the Green potential of a function \(f(x)\) on an open connected and bounded set \(\Omega\) in \(\mathbb{R}^n\). The paper is devoted to the extension of the inequality \[ \Biggl( \int_\Omega |\nabla Gf|^q dx\Biggr)^{1/q}\leq c(\Omega,p) \Biggl( \int_\Omega |f|^p dx\Biggr)^{1/p} \tag{1} \] with \(1/q= 1/p-1/n\), \(n/(n-1)< q< \infty\), well known in case of a sufficiently smooth boundary \(\partial\Omega\), to the case of a “highly non-rectifiable” boundary (the case of a Lipschitzian boundary was treated by B. Dahlberg [Math. Scand. 44, 149-170 (1979; Zbl 0418.31003)]). The restrictions on \(\Omega\) are given in terms of the so called non-tangentially accessible (NTA) domains, this notion being developed by D. S. Jerison and C. E. Kenig [Adv. Math. 46, 80-147 (1982; Zbl 0514.31003)]. The restrictions on the exponent \(q\) for (1) to be valid are given in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.
Reviewer: S.G.Samko (Faro)

MSC:

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
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