On Limits at Infinity of Weighted Sobolev Functions

TL;DR

This paper characterizes conditions on Muckenhoupt weights that ensure Sobolev functions have well-defined limits at infinity in radial and vertical directions, extending previous results with new proofs and sharp conditions.

Contribution

It provides a complete characterization of limit existence for weighted Sobolev functions under Muckenhoupt weights, including sharp conditions and new proofs for classical results.

Findings

Radial limits exist under specific weight conditions.

Vertical limits have sharp sufficient conditions.

Results generalize and improve upon Fefferman and Uspenski
2f's work.

Abstract

We study necessary and sufficient conditions for a Muckenhoupt 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 $|\nabla u|\in L^p(\mathbb R^d,w)$. 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 Uspenski\u{\i}.