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Boundary limits of monotone Sobolev functions in Musielak-Orlicz spaces on uniform domains in a metric space. (English) Zbl 1376.30045
Let $$(X,d,\mu)$$ be a metric measure space with a measure $$\mu$$ satisfying the doubling condition; $$D\subset X$$ is a uniform domain, which essentially means that we can join any two points $$x,x'\in D$$ by a cigar-shaped set which is not too thin and not too crooked. The authors consider functions $$u$$ in a variable-order Musielak-Orlicz space on $$D$$ satisfying the condition that for some positive $$g\in L^{p_0}_{\text{loc}}$$ $|u(x)-u(x')| \leq C r\left(\mu(B(y,\sigma r))^{-1} \int_{B(y,\sigma r)} g(z)^{p_0}\,\mu(dz)\right)^{1/p_0}$ for all $$x,x'\in B(y,r)$$ with $$B(y,\sigma r)\subset D$$ for some $$\sigma>1$$ and $\int_D \Phi(z,g(z))\delta_D(z)^\alpha\,\mu(dz)<1.$ Here, $$\delta_D(z)$$ is the distance of $$z$$ to the boundary, $$\Phi(z,t)= t^{p(z)}\phi(z,t) \mathbf{1}_{(0,\infty)}(t)$$ (with $$\phi$$ satisfying certain doubling and log-Hölder conditions), and $$0 < p_0 \leq \inf_X p$$. The main theorem exhibits a set $$E_{\beta,s}\subset \partial D$$ with the following property: if $$s+\alpha-1<\inf_X p \leq \sup_X p < s+\alpha$$ all points $$\xi\in\partial D\setminus E_{\beta,s}$$ such that there is a rectifiable curve inside $$D$$ tending to $$\xi$$ along which $$u$$ has a finite limit $$L$$ admit $$L$$ as tangential limit of order $$\beta$$.
This paper extends earlier results by the present authors and T. Futamura [Rev. Mat. Complut. 28, No. 1, 31–48 (2015; Zbl 1308.30066)] pushing the technique developed by F. Di Biase et al. [Ill. J. Math. 57, No. 4, 1025–1033 (2013; Zbl 1309.31007)] through to Musielak-Orlicz spaces.

##### MSC:
 30L99 Analysis on metric spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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