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Generalized Cranston-McConnell inequalities for discontinuous superharmonic functions. (English) Zbl 0901.31001

Let \(\Omega\subseteq \mathbb{R}^2\) be an open set with finite area, and let \(G(\cdot,\cdot)\) denote its Green function. Further, if \(n\in \mathbb{N}\) and \(\eta>1\), let \(\Phi\) be a non-negative Borel measurable function on \((0,\infty]^n\) and \(\Psi_\eta (t_1,\dots, t_n)= \sup\Phi (c_1t_1,\dots, c_nt_n)\), where the supremum is over all the \((c_1,\dots, c_n)\) in \((\eta^{-2},\eta^2)^n\). The main result of this paper asserts that, if \(u,v_1,\dots,v_n\) are positive superharmonic functions on \(\Omega\), then the function \[ (1/u) \int_\Omega G(\cdot,y) u(y) \Phi(v_1(y),\dots,v_n(y))dy \] is bounded above on \(\Omega\) by \(C\int_\Omega \Psi_\eta (v_1(y),\dots,v_n(y))dy\), where \(C\) is a positive constant depending only on \(n\) and \(\eta\). The special case where \(\Phi\equiv 1\) and \(u\) is harmonic on \(\Omega\) is due to M. Cranston and T. R. McConnell [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 1-11 (1983; Zbl 0506.60071)]. This was then extended to general \(\Phi\) and continuous superharmonic functions \(u,v_1,\dots,v_n\) by the author and M. Murata [J. Anal. Math. 69, 137-152 (1996; Zbl 0865.31009)]. The present paper removes the assumption of continuity by means of fine potential theory.

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31C40 Fine potential theory; fine properties of sets and functions
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