Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?

Question

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?

Answer 1

You may want to take a look at

• Herbert Koch and Winfried Sickel, "Pointwise multipliers of Besov spaces of smoothness zero and spaces of continuous functions", Rev. Mat. Iberoamericana 18 (2002), 587–626.

They established that the set of all distributions $f$ such that $g \mapsto fg$ is a bounded linear map from $B^0_{\infty,1}$ to itself is a strict subspace of $B^0_{\infty,1}$. Based on their result I think Triebel made an error there.

In particular, see their Remark 15, which states a variant of what I wrote in my first comment (that a sufficient condition is for $f\in B^0_{\infty,1}$ and that $\sup_{j \in \mathbb{N}} j \|P_j f|_\infty < \infty$).

The explicit function given in part (i) of their Lemma 16 should be a function $f\in B^0_{\infty,1}$ such that $f^2 \not\in B^0_{\infty,1}$.