Is the product of two Sobolev functions in $L^p$?

Question

Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$.

Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$.

Under what conditions on $\beta$ can we get that $fg\in L^p(R^n)$?

Many thanks for the answer!

Answer 1

No HOPE.

Consider $g_M\in C^\infty_0(\mathbb R)$, such that, $0\le g_M(x)\le 1$, for all $x\in\mathbb R$, $g_M(x)=0$, for $|x|>M+1$ and $g_M(x)=1$, for $|x|\le M$.

Then $g_M\in W^{\beta,p}$ for all $\beta\in\mathbb R$ and $p\ge 1$.

Meanwhile, $g_Mf\to f$ in the sence of $W^{\alpha-1,p}-$norm, and hence, for $M$ large enough, $g_Mf\not\in L^p(\mathbb R^n)$.