  This paper is devoted to studying the local well-posedness (existence,
uniqueness and continuous dependence) for the Camassa-Holm type equations in
critial Besov spaces $B^{1+\frac{1}{p}}_{p,1},1\leq p<+\infty$, which improves
the past index $s> \max\{\frac{3}{2},1+\frac{1}{p}\}$ or $s=1+\frac{1}{p},\
p\in[1,2],\ r=1$ in \cite{d1,d2,liy}. The proof is based on the Lagrangian
coordinate transformation. Indeed, we first establish the Lagrangian
coordinates and obtain the $W^{1,\infty}\cap W^{1,p}$ norm estimates for the
solutions in the Lagrangian coordinates. Then, returning to the original
coordinates, we get the $L^{p}$ norm estimates for the solutions in the
original coordinates. Finally, combining with the interpolation inequalities,
we prove the uniqueness and continuous dependence of the Camassa-Holm type
equations in critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},1\leq p<+\infty$.
