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

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?

• I agree with your analysis, but the outlined method is not quite sharp. If you go a few lines up, you see that for estimating the $\Sigma'''$ terms what you need is to estimate $\sum_{k = 1}^\infty \sum_{l= k}^\infty \|b_l\|_{\infty} \|c_l\|_{\infty}$. This sum is equal to $\sum_{k = 1}^\infty k \|b_k\|_\infty \|c_k\|_\infty$. I don't see quite how to bound this by $\|b\|_{\ell_1(L^\infty)} \|c\|_{\ell_1(L^\infty)}$ yet, but this looks more plausible. (For example, this is clearly true if $b$ or $c$ is monotone. Commented Apr 15, 2021 at 1:37
• @WillieWong Thanks for your comment. Indeed, it's not clear to me either. Would you please clarify what you mean by "monotone"? Commented Apr 15, 2021 at 1:47
• If $\|b_k\|_{\infty}$ is monotonically decreasing, then $\sup_k k \|b_k\|_\infty \leq \|b\|_{\ell_1(L^\infty)}$. This is not true for non-monotone sequences. Commented Apr 15, 2021 at 1:48

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 $$f^2 \not\in B^0_{\infty,1}$$.

• Thank you so much for sharing this article! I will take a look at it. Commented Apr 15, 2021 at 2:50
• It seems this question, with an acknowledgement of Triebel's error, was taken up in this article by Sickel and Triebel. See the paragraph at the top of pg. 12. Commented Apr 15, 2021 at 16:18
• Yeah, if I weren't still on my year-long "work from home", the first thing I'd have done would be to go to my bookshelf in my office and pull down my copies of Triebel's "Theory of Functional Spaces" to check if this was clarified. Commented Apr 15, 2021 at 16:30