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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
##### Keywords:
nontangential limits; uniform domains