A1 Journal article (refereed)
On Limits at Infinity of Weighted Sobolev Functions (2022)

Eriksson-Bique, S., Koskela, P., & Nguyen, K. (2022). On Limits at Infinity of Weighted Sobolev Functions. Journal of Functional Analysis, 283(10), Article 109672. https://doi.org/10.1016/j.jfa.2022.109672

JYU authors or editors

Koskela, Pekka
Nguyen, Khanh

Publication details

All authors or editors: Eriksson-Bique, Sylvester; Koskela, Pekka; Nguyen, Khanh

Journal or series: Journal of Functional Analysis

ISSN: 0022-1236

eISSN: 1096-0783

Publication year: 2022

Volume: 283

Issue number: 10

Article number: 109672

Publisher: Elsevier

Publication country: Belgium

Publication language: English

DOI: https://doi.org/10.1016/j.jfa.2022.109672

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/82817

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2201.10876v1

Additional information: Dedicated to Professor Olli Martio on the occasion of his 80th birthday celebration.

Abstract

We study necessary and sufficient conditions for a Muckenhoupt weight w∈Lloc1(Rd) that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions u∈Wloc1,p(Rd,w) with a p-integrable gradient |∇u|∈Lp(Rd,w) where 1≤p<∞ and 2≤d<∞. 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<∞), which depend on the integrability degree of its gradient.

Keywords: mathematics; differential equations; functions (mathematical methods)

Free keywords: Sobolev functions; Muckenhoupt; limit; asymptotic

Fields of science: