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Limits at infinity for functions in fractional Sobolev spaces

Angha Agarwal, Pekka Koskela, Kaushik Mohanta

TL;DR

This paper analyzes the asymptotic pointwise behavior at infinity for functions in the homogeneous fractional Sobolev spaces ${\dot W}^{s,p}_q({\mathbb R}^n)$. It proves an optimal limit-at-infinity theorem: for $0<s<1$ and $sp<n$, each $u\in{\dot W}^{s,p}_q({\mathbb R}^n)$ has a unique asymptotic value $K_{u}$ outside a $(p,s)$-thin-at-infinity set, enabling almost-everywhere radial and vertical limits. The results generalize classical radial/vertical limit phenomena to nonlocal, fractional settings and reveal a rigidity: when $q\ge\frac{np}{n-sp}$ the space collapses to constants. The work also provides sharp counterexamples and delineates the precise parameter regimes where limits may fail, informing the structure of fractional Sobolev spaces at infinity.

Abstract

We establish optimal results on limits at infinity for functions in fractional Sobolev spaces.

Limits at infinity for functions in fractional Sobolev spaces

TL;DR

This paper analyzes the asymptotic pointwise behavior at infinity for functions in the homogeneous fractional Sobolev spaces . It proves an optimal limit-at-infinity theorem: for and , each has a unique asymptotic value outside a -thin-at-infinity set, enabling almost-everywhere radial and vertical limits. The results generalize classical radial/vertical limit phenomena to nonlocal, fractional settings and reveal a rigidity: when the space collapses to constants. The work also provides sharp counterexamples and delineates the precise parameter regimes where limits may fail, informing the structure of fractional Sobolev spaces at infinity.

Abstract

We establish optimal results on limits at infinity for functions in fractional Sobolev spaces.
Paper Structure (6 sections, 23 theorems, 136 equations)

This paper contains 6 sections, 23 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Counterexamples
  4. Proof of Theorem \ref{['ps-thin theorem']}
  5. The case of radial limits
  6. The case of vertical limits

Key Result

Theorem 1.1

Let $u^*\in\dot{W}^{s,p}_q({\mathbb R}^n)$ with $0<s<1$, $0<p,q<\infty$ and $sp<n$. Then there exists a set $E$ which is $(p,s)$-thin at infinity and $K_u\in{\mathbb R}$ so that

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 41 more