We consider the discrete LlogL space of sequences $x=(x_i)$ which contains all sequences for which the following norm is finite

$$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$

Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL space.

Can we estimate for sequences $y_i, x_i \ge 0$ both summing up to one in terms of some function $C(t,s)$ depending only on $t$ and $s$ the Lipschitz estimate

$$\Big \vert \sum_i x_i\log(x_i)-y_i \log(y_i) \Big \vert \le C(\Vert x\Vert_{LL},\Vert y\Vert_{LL}) \Vert x-y\Vert_{\ell^1} $$