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Compact Sobolev embeddings of radially symmetric functions

Zdeněk Mihula

TL;DR

This work resolves the compactness problem for Sobolev embeddings of radially symmetric functions on $\mathbb{R}^n$ within the broad setting of rearrangement-invariant spaces, extending to higher-order Sobolev spaces and incorporating weighted embeddings on balls. The authors introduce a reduction principle via a one-dimensional Hardy/Copson-type operator and define the optimal target ri-space $Y_X$ to capture the smallest feasible target in $W^{m}_R{X}(B_R)\hookrightarrow Y(B_R, d\mu_{\alpha})$. A key advance is a tail-decay analysis for radial functions on unbounded domains, enabling a complete characterization of compact embeddings on $\mathbb{R}^n$ and, separately, on balls with weights, including sharp criteria for Lorentz–Zygmund spaces. The results unify and sharpen previous partial compactness results, provide explicit conditions in terms of fundamental functions and Lorentz–Zygmund parameters, and yield concrete applications to nonlinear PDE analysis where radial symmetry mitigates loss of compactness at infinity. Overall, the paper delivers a comprehensive, sharp framework for understanding compact Sobolev embeddings in the radial, ri-space setting with broad implications for unbounded-domain problems.

Abstract

We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space $\mathbb{R}^n$ in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary restrictions and cover also embeddings of higher order, providing a complete picture within this framework. To achieve this, we need to develop new techniques because the usual techniques used in the study of compactness of Sobolev embeddings in the general framework of rearrangement-invariant function spaces are limited to domains of finite measure, which is essential for them to work. Furthermore, we also study certain weighted Sobolev embeddings of radially symmetric functions on balls. We completely characterize their compactness and also describe optimal target rearrangement-invariant function spaces in these weighted Sobolev embeddings.

Compact Sobolev embeddings of radially symmetric functions

TL;DR

This work resolves the compactness problem for Sobolev embeddings of radially symmetric functions on within the broad setting of rearrangement-invariant spaces, extending to higher-order Sobolev spaces and incorporating weighted embeddings on balls. The authors introduce a reduction principle via a one-dimensional Hardy/Copson-type operator and define the optimal target ri-space to capture the smallest feasible target in . A key advance is a tail-decay analysis for radial functions on unbounded domains, enabling a complete characterization of compact embeddings on and, separately, on balls with weights, including sharp criteria for Lorentz–Zygmund spaces. The results unify and sharpen previous partial compactness results, provide explicit conditions in terms of fundamental functions and Lorentz–Zygmund parameters, and yield concrete applications to nonlinear PDE analysis where radial symmetry mitigates loss of compactness at infinity. Overall, the paper delivers a comprehensive, sharp framework for understanding compact Sobolev embeddings in the radial, ri-space setting with broad implications for unbounded-domain problems.

Abstract

We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary restrictions and cover also embeddings of higher order, providing a complete picture within this framework. To achieve this, we need to develop new techniques because the usual techniques used in the study of compactness of Sobolev embeddings in the general framework of rearrangement-invariant function spaces are limited to domains of finite measure, which is essential for them to work. Furthermore, we also study certain weighted Sobolev embeddings of radially symmetric functions on balls. We completely characterize their compactness and also describe optimal target rearrangement-invariant function spaces in these weighted Sobolev embeddings.

Paper Structure

This paper contains 12 sections, 16 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rearrangements and r.i. spaces
  4. Sobolev spaces
  5. Sobolev functions with radial symmetry on the entire space
  6. Weighted embeddings on balls
  7. Optimal target space in the weighted Sobolev embedding
  8. Compactness of the weighted Sobolev embedding
  9. Proof of Theorem \ref{['thm:characterization_of_comp_emb_Y']}
  10. Examples
  11. Sobolev embeddings on the entire space
  12. Weighted Sobolev embeddings on balls

Key Result

Theorem 1.1

Let $m,n\in\mathbb{N}$, $n\geq2$. Let $X(\mathbb{R}^n)$ and $Y(\mathbb{R}^n)$ be rearrangement-invariant function spaces. The Sobolev embedding is compact if and only if and simultaneously either and one of the conditions is satisfied or and one of the conditions is satisfied.

Theorems & Definitions (38)

  • Theorem 1.1
  • Proposition 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.1
  • proof : Proof of \ref{['prop:bounded_set_in_WmrX_tail_is_small']}
  • Proposition 3.3
  • proof
  • ...and 28 more