Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm $$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$ where $\{P_j\}_{j=0}^\infty$ is an inhomogeneous Littlewood-Paley partition of unity with the convention that $P_0$ is the low-frequency projector. I am interested in when $B_{p,q}^s$ is a multiplication algebra: $$\|fg\|_{B_{p,q}^s} \lesssim \|f\|_{B_{p,q}^s} \|g\|_{B_{p,q}^s}.$$

It is well-known (for example, see Corollary 2.86 in Bahouri, Chemin, and Danchin's *Fourier Analysis and Nonlinear Partial Differential Equations*) that if $s>0$, then $L^\infty \cap B_{p,q}^s$ is an algebra with
$$\|fg\|_{B_{p,q}^s} \lesssim_s \|f\|_{L^\infty}\|g\|_{B_{p,q}^s} + \|g\|_{L^\infty}\|f\|_{B_{p,q}^s}.$$
Note that this result does not cover the case $B_{\infty,1}^0$. In the following article, Triebel claims (see Theorem 2) that $B_{\infty,1}^0$ is a multiplication. But if you look at the proof (the last block of equations before Remark 3 on pg. 41), there seems to be error in obtaining the second line given that
$$\sum_{k=1}^\infty\sum_{l=k}^\infty 2^{ksq} \|b_l\|_{L^p}^q \neq \sum_{l=1}^\infty 2^{lsq} \|b_l\|_{L^p}^q$$
if $s=0$. So my question is the following:

Question.Is the space $B_{\infty,1}^0$ a multiplication algebra?