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On the modulus of continuity of fractional Orlicz-Sobolev functions

Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková

Abstract

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $\rn$ to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

On the modulus of continuity of fractional Orlicz-Sobolev functions

Abstract

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.
Paper Structure (14 sections, 13 theorems, 260 equations)

This paper contains 14 sections, 13 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Young functions
  3. Function spaces
  4. Spaces of continuous functions
  5. Orlicz and Orlicz-Lorentz spaces
  6. Rearrangement-invariant spaces
  7. Integer-order Sobolev spaces
  8. Fractional Orlicz-Sobolev spaces
  9. Main results
  10. Proofs of the main results
  11. Proof of Theorem \ref{['necessity']}
  12. Proof of Theorem \ref{['thm:s<1']}: case $s\in (0,1)$
  13. Proof of Theorem \ref{['thm:1<s<n']}: case $s\in (1,n)$
  14. Proof of Theorem \ref{['thm:n<s<n+1']}: case $s\in (n,n+1)$

Key Result

Proposition 2.1

Let $s\in (0,n)$ and let $A$ be a Young function. Assume that condition convinf is fulfilled. Let $E$ and $\vartheta_s$ be defined as above. Then: (i) $E$ is a Young function. (ii) We have that (iii) The function $\vartheta_s$ is non-decreasing. (iv) Let $B$ be the Young function given by Then, (v) If then $B\simeq A$ near infinity, and (vi) If then $B\simeq A$ near $0$, and

Theorems & Definitions (24)

  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:E']}
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['prop:F']}
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • ...and 14 more