For $f\in\mathcal{S}$ a Schwartz function on $\mathbb{R}^n$ and $m$ a bounded function, define $Tf$ by $\widehat{T f}=m\cdot \widehat{f}$.

Fix $1<p<\infty$, $p\not=2$. Suppose we have proved that there exists a constant $C>0$ such that $$ \|Tf\|_p \le C \|f\|_p$$ holds for all $f\in\mathcal{S}$ satisfying $\widehat{f}\ge 0$.

Is it still possible that $T$ is not $L^p\to L^p$ bounded?

I believe that the answer is yes, but I don't know of an example. I'd also be grateful for any reference where this or similar situations (possibly in the context of other kinds of operators) are discussed.