The jump problem for the critical Besov space

We introduce the critical Besov space \( B_{p} \) $B_p$ on quasicircles and prove the solvability of the jump problem on d-regular quasicircles with \( B_{p} \) $B_p$ boundary values for \( d<p<\frac{d}{d-1}\) $d<p<\frac{d}{d-1}$ . As applications, we obtain the \( B_{p}\, (1<p<\infty ) \) $B_p(1<p<\infty )$ boundedness of the Cauchy integral operator and the Faber operator on chord-arc curves.