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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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