## Abstract

In this paper we analyse functions in Besov spaces Bq,∞1/q(RN,Rd),q∈(1,∞) , and functions in fractional Sobolev spaces W^{r}^{,}^{q}(R^{N}, R^{d}) , r∈ (0 , 1) , q∈ [1 , ∞) . We prove for Besov functions u∈Bq,∞1/q(RN,Rd) the summability of the difference between one-sided approximate limits in power q, | u^{+}- u^{-}| ^{q} , along the jump set J_{u} of u with respect to Hausdorff measure H^{N}^{-}^{1} , and establish the best bound from above on the integral ∫Ju|u+-u-|qdHN-1 in terms of Besov constants. We show for functions u∈Bq,∞1/q(RN,Rd),q∈(1,∞) that lim infε→0+1εN∫Bε(x)|u(z)-uBε(x)|qdz=0 for every x outside of a H^{N}^{-}^{1} -sigma finite set. For fractional Sobolev functions u∈ W^{r}^{,}^{q}(R^{N}, R^{d}) we prove that limε→0+1εN∫Bε(x)1εN∫Bε(x)|u(z)-u(y)|qdzdy=0 for H^{N}^{-}^{r}^{q} a.e. x, where q∈ [1 , ∞) , r∈ (0 , 1) and rq≤ N . We prove for u∈ W^{1}^{,}^{q}(R^{N}) , 1 < q≤ N , that limε→0+1εN∫Bε(x)|u(z)-uBε(x)|qdz=0 for H^{N}^{-}^{q} a.e. x∈ R^{N} . In addition, we prove Lusin-type approximation for fractional Sobolev functions u∈ W^{r}^{,}^{q}(R^{N}, R^{d}) by Hölder continuous functions in C^{,}^{r}(R^{N}, R^{d}) .

Original language | English |
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Article number | 28 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 63 |

Issue number | 2 |

DOIs | |

State | Published - 1 Mar 2024 |

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics