Summary: We study necessary and sufficient conditions for a Muckenhoupt \(\mathcal{A}_p\)-weight \(w \in L_{\operatorname{loc}}^1(\mathbb{R}^d)\) that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions \(u \in W_{\operatorname{loc}}^{1, p}(\mathbb{R}^d, w)\) with a \(p\)-integrable gradient \(| \nabla u | \in L^p(\mathbb{R}^d, w)\) where \(1 \leq p < \infty\) and \(2 \leq d < \infty\). The question is shown to subtly depend on the sense in which the limit is taken.
First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of C. Fefferman [Ann. Inst. Fourier 24, No. 3, 159–164 (1974; Zbl 0292.26013)] and S. V. Uspenskij [Transl., Ser. 2, Am. Math. Soc. 87, 121–145 (1961; Zbl 0198.46106); translation from Trudy Mat. Inst. Steklov 60, 282–303 (1961)].
As applications to partial differential equations, we give results on the limiting behavior of weighted \(q\)-harmonic functions at infinity \((1 < q < \infty)\), which depend on the integrability degree of its gradient.