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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 \(\xi\).

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