Approximations in Besov Spaces and Jump Detection of Besov Functions with Bounded Variation

In this paper, we provide a proof that functions belonging to Besov spaces Bʳq,∞(RN,Rᵈ), q∈ [1,∞), r∈(0,1), satisfy the following formula under a certain condition: {equation} {eq:main result in abstract} lim_{{ε}→ 0^+}{1}{|ln{ε}|}[uε]^q_{Wr,q(R^N,R^d)}=Nlim_{{ε}→ 0^+}∫R^N{1}{{ε}^N}∫_{Bε(x)}{|u(x)-u(y)|^q}{|x-y|rq}dydx. {equation} Here, [·]_Wr,q represents the Gagliardo seminorm, and uε denotes the convolution of u with a mollifier η(ε)(x):=1/εNη(x/ε), η∈ W1,1(RN),∫RNη(z)dz=1. Furthermore, we prove that every function u in BV(RN,Rᵈ)∩ B1/pp,∞(RN,Rᵈ),p∈(1,∞), satisfies {multline} limε→ 0^+{1}{|ln{ε}|}[uε]^q_{W1/q,q(R^N,R^d)}=Nlim_{{ε}→ 0^+}∫R^N{1}{{ε}^N}∫_{Bε(x)}|u(x)-u(y)|^q/|x-y|dydx =(∫_{SN-1}|z_1|~dHN-1(z))∫J_u |u^+(x)-u^-(x)|^q dHN-1(x), {multline} for every $1<q<p$. Here u⁺,u⁻ are the one-sided approximate limits of u along the jump set Jᵤ.