Let \(D\subset\mathbb{R}^n,\) \(n\geq 2\) denote the upper half space, i.e., \(D=\{x_n>0\}.\) A continuous function \(u\) on \(D\) is said to be monotone in the sense of Lebesgue if for every relatively compact open subset \(G\) of \(D,\) the following equalities hold: \(\max_{\overline{G}}u=\max_{\partial G}u\) and \(\min_{\overline{G}}u=\min_{\partial G}u.\)
The authors extend the previous results obtained in [Y. Mizuta, Ann. Acad. Sci. Fenn., Ser. A I, Math. 20, 315–326 (1995; Zbl 0852.31008) and J. J. Manfredi and E. Villamor, Ill. J. Math. 45, 403–422 (2001; Zbl 0988.30015)]. The main result is the following theorem. Let \(\mu\) be a Borel measure on \(\mathbb{R}^n\) satisfying the doubling condition. Denote \(u_B:=\frac{1}{B}\int_Bu(y) \,d\mu(y).\) Let \(u\) be a monotone function on \(D\) with \(g\) satisfying \[ | u(x)-u_B| \leq Mr\left(\frac{1}{\mu(\sigma B)}\int_{\sigma B}g(z)^p \,d\mu(z)\right)^{1/ p} \] and \[ \int_Dg(z)^p \,d\mu(z)<\infty \] for every \(x\in B\) with \(\sigma B\subset D,\) where \(\sigma>1\) and \(B=B(y,r).\) Suppose \(p>s-1,\) and set \[ E=\left\{\xi\in\partial D:\limsup_{r\to 0}(r^{-p}\mu(B(\xi,r)))^{-1}\int_{B(\xi,r)\cap D}g(z)^p \,d\mu(z)>0\right\}. \] 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, then \(u\) has a nontangential limit at \(\xi.\)