[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O]
Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity.
My Questions:
(I) Can we expect to choose $0\neq g \in C_{c}^{\infty}(\mathbb R)$ (may be depending on given $f$) so that $$\|(g\widehat{(f^{3}})^{\vee}\|_{L^{p}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{s}}$$ for some $1\leq p \leq 2,$ and for some $r \geq 1, s\geq p$? (where $C$ is some constant)\
(II) Or, Can we expect to choose $0 \neq g= g_{1}g_{2} \in C_{c}^{\infty}(\mathbb R)$ with $g_{1}, g_{2} \in C_{c}^{\infty}(\mathbb R)$ with $g_{1}\leq 1$ so that $$\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$$ for some $1\leq p \leq 2,$ and for some $r \geq 1, s\geq p$?
[where $\widehat{(f^{3})}$ denotes the Fourier transform of $f^{3}$ and $(g\widehat{(f^{3})})^{\vee}$ denotes the inverse Fourier transform of $g\widehat{(f^{3})}$; and $ \ C_{c}(\mathbb R)$ is the class of smooth functions with compact support] [Side remark: see the related question]
Any ideas or comments would be appreciated. Thanks,
