Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions

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-rq 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) .