  For the famous Camassa-Holm equation, the well-posedness in
$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p\leq2$ and the ill-posedness
in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $1\leq p\leq+\infty,\
1<r\leq+\infty$ had been studied in \cite{d1,d2,glmy}. That is to say, it left
an open problem in the critical case $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with
$2<p\leq+\infty$ proposed by Danchin in \cite{d1,d2}. In this paper, we solve
this problem. The main difficulty is to prove the uniqueness, which usually
needs to use the Moser-type inequality, resulting in the index $p$ belongs to
$[1,2]$. To overcome the difficulty, inspired by Linares, Ponce and Thomas
\cite{lps}, we combine the Lagrange coordinate transformation and small time
conditions to avoid using the Moser-type inequality. As a result, we obtain the
local well-posedness for the Camassa-Holm equation in critical Besov spaces
$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p<+\infty$. It is worth
mentioning that our method is suitable for many Camassa-Holm type equations
such as the Novikov equation and the two-component Camassa-Holm system, which
can also improve their index on the local well-posedness.
