I have a problem of an estimate of $(1.3)$ in the paper https://arxiv.org/abs/2010.10460, which says that
Assume $$\mathcal{F}\{Q_m[f,g]\}(\xi)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbb{R}^3}m(\xi,\eta)\hat{f}(\xi-\eta)\hat{g}(\eta)d\eta,$$ then we have using a variant of Holder's inequality $$||Q_m[f,g]||_{L^r}\lesssim||\mathcal{F}m||_{L^1}||f||_{L^p}||g||_{L^q},\quad\quad \frac{1}{r}=\frac{1}{p}+\frac{1}{q}.$$ I know this variant of Holder's inequality Suppose $1\leq r\leq \infty,r\leq q\leq\infty,r\leq p\leq \infty,$ such that $$\frac{1}{r}=\frac{1}{p}+\frac{1}{q},$$ and let $u\in L^p(\Omega), v\in L^q(\Omega),$ then $$||uv||_{L^r}\leq ||u||_{L^p}||v||_{L^q}.$$ I tried to estimate $||\mathcal{F}\{Q_m[f,g]\}(\xi)||_{L^r}$ using Holder's inequality but I can't get the author's estimate. How can I get the target estimate from the Fourier space to the Euclidean space?