Eriksson-Bique, S., Koskela, P., & Nguyen, K. (2022). On limits at infinity of weighted Sobolev functions. Journal of Functional Analysis, 283(10), 109672. https://doi.org/10.1016/j.jfa.2022.109672

### On limits at infinity of weighted Sobolev functions

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Author: Eriksson-Bique, Sylvester1; Koskela, Pekka2; Nguyen, Khanh2
Organizations: 1Research Unit of Mathematical Sciences, P.O. Box 8000, FI-90014 Oulu, Finland
2Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.7 MB)
Language: English
Published: Elsevier, 2022
Publish Date: 2022-12-28
Description:

# Abstract

We study necessary and sufficient conditions for a Muckenhoupt $$\mathscr{A}_p$$-weight $$w \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $$u \in W^{1,~p}_{\mathrm{loc}}(\mathbb{R}^{d}, w)$$ with a $$p$$-integrable gradient $$|\nabla u| \in L^{p}(\mathbb{R}^{d}, w)$$ where $$1 \le p \lt \infty$$ and $$2 \le d \lt \infty$$. 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 Fefferman and Uspenskiĭ.

As applications to partial differential equations, we give results on the limiting behavior of weighted $$q$$-harmonic functions at infinity ($$1 \lt q \lt \infty$$ ), which depend on the integrability degree of its gradient.

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Series: Journal of functional analysis
ISSN: 0022-1236
ISSN-E: 1096-0783
ISSN-L: 0022-1236
Volume: 283
Issue: 10
Article number: 109672
DOI: 10.1016/j.jfa.2022.109672