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Lindelöf theorems for monotone Sobolev functions on uniform domains. (English) Zbl 1070.31001
Let \((X,d)\) be a metric space with a Borel measure \(\mu\) with \(\mu(B(z,r)) \in (0, \infty)\) for all \(z \in X, r >0,\) where \(B(z,r)\) stands for the ball centered at \(z \in X \) and with radius \(r>0.\) Assume further that there is a constant \(C_1\geq 1\) such that \(\mu(2B) \leq C_1 \mu(B)\) holds for all balls \(B\) and that there are constants \(Q \geq 1, C_2 > 0,\) with \[ \mu(B) /\mu(B_0) \geq C_2 (\text{diam}( B) / \text{diam} (B_0))^Q \] for all balls \(B, B_0\) with \(B \subset B_0.\) Let \(D \subset X,\) consider functions \(u : D \to \mathbb R\) for which there exists a nonnegative Borel measurable \(g \in L^p(D; \mu)\), \(p>1\), \(M >0\), \(\lambda \in (0,1],\) with \[ | u(x)-u(x')| \leq M r \left( {1 \over \mu(B(x,r))} \int_{B(x,r)} g(y)^p d \mu(y)\right)^{1/p} \] for \(x,x' \in B(z, \lambda r)\) whenever \(B(x,r) \subset D\) and \[ \int_D g(y)^p \delta_D(y)^\alpha \, d \mu(y) < \infty \,, \] \(\delta_D(z) = d(z, \partial D)\). Call functions satisfying these conditions functions of class \({\mathcal U}(D)\). The author proves the following result.
Theorem. Let \(D\) be a uniform domain in \(X,\) \(u \in {\mathcal U}(D) \), and for \(p > Q +\alpha-1\) set \[ E= \biggl\{ \xi \in \partial D : \lim \sup_{r \to 0} {r^{p-\alpha} \over \mu(B(\xi,r))} \int_{B(\xi,r) \cap D} g(y)^p \delta_D(y)^{\alpha} d \mu(y) > 0 \biggr\}. \] If \(\xi \in \partial D \setminus E\) and there exists a curve \(\gamma \) in \(D\) tending to \(\xi\) along which \(u\) has a finite limit \(L\), then \(u\) has nontangential limit \(L\) at \(\xi\).

MSC:
31B25 Boundary behavior of harmonic functions in higher dimensions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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