I found the following lemma in some old paper
with proof referred in unavailable reference
I am trying to check why point (iii) is true. I guess the constants $C_0, C_1$ were exchanged.
I tried to proceed as follows: Since $a\in L^1(\mathbb{R}^n) \cap L^\infty(\mathbb{R}^n),$ we have $a\in L^q(\mathbb{R}^n)$ for any $1\leq q\leq \infty.$ Particularly, we take $q = \frac{p}{2-p}.$ Then, using Hausdorff–Young inequality, we get $$\| \mathcal{F}^{-1}(a \hat v) \|_{p'} \leq \|a \hat v\|_p.$$ Using Hölder's inequality together with Hausdorff–Young inequality, we get $$\| \mathcal{F}^{-1}(a \hat v) \|_{p'} \leq \|a\|_q\| \hat v\|_{p'}\leq \|a\|_q\| v\|_{p}.$$
By interpolation on $a,$ we find that $$\|a\|_q \leq \|a\|_\infty^{1-\theta} \|a\|_1^{\theta} \leq C_1^{1-\theta} \|a\|_1^{\theta}, \quad \theta = \frac 1q= \frac2p -1.$$
Is assumption (i) written incorrectly ? so that the author meant $\|a\|_1 \leq C_0$, or there is another proof.
Thank you
You're taking a loss by using Holder. Defining $T(v) = \mathcal{F}^{-1}(a\widehat{v})$, (i) gives you $$ \|T\|_{L^1 \to L^\infty} \leq C_0 $$ and (ii) gives you $$ \|T\|_{L^2 \to L^2} \leq C_1. $$ Applying the Riesz-Thorin interpolation theorem directly to $T$ yields the desired conclusion.