Bound for the product of Sobolev functions in $W^{s,1}$

Question

I would like to bound the product of two functions $f$, $g$ in the space $W^{s,1}$.

$$ \lVert fg\rVert_{W^{s,1}}\leq c\lVert f \rVert \lVert g \rVert. $$ It seems reasonable to want to use Hölder's inequality and have the norms given by it on the right-hand side, however I am not sure whether the operator $$J_s=(1-\Delta)^{s/2}$$

is bounded in $L^1$, as this is a borderline case of the Calderon-Zygmund inequality.

More generally, is $W^{s,1}$ a Banach algebra? If yes, for what $s$, $d$?

Answer 1

I believe that $W^{s,1}(\mathbb{R}^d)$ is a Banach algebra when $s > d$.

This is a particular case of Theorem 7.3 of the paper Multiplication in Sobolev spaces by Ali Behzadan and Michael Holst.