I am reading "Estimates for translation-invariant operators" by Lars Hormander. Currently, I am stuck at the following argument which seems obvious to Lars ))). Let $\varphi(\xi)$ be such a function that the following inequalities hold $$ \left(\int\limits_{\mathbb{R}^n}\left|\frac{\widehat{f}}{\varphi}\right|^p\varphi^2\,d\xi\right)^{\frac1p}\lesssim \|f\|_{L_p(\mathbb{R}^n)},\quad 1<p\leq 2.\\ $$ If we combine this inequality with Hausdorff-Young inequality $$ \|\widehat{f}\|_{L_{p'}(\mathbb{R})^n}\lesssim \|f\|_{L_p(\mathbb{R}^n)},\quad 1\leq p \leq 2 $$ and use Holder inequality, then it is possible to obtain the following estimate $$ \left(\int\limits_{\mathbb{R}^n}\left|\widehat{f}\varphi^{\frac1r-\frac1{p'}}\right|^r\,d\xi\right)^{\frac1r}\lesssim \|f\|_{L_p(\mathbb{R}^n)},\quad 1\leq p \leq r \leq p'. $$ Why is it true? Please, help me.
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