Can anyone give me some references to prove this statement that I found in
Hyakuna, Ryosuke; Tsutsumi, Masayoshi, On existence of global solutions of Schrödinger equations with subcritical nonlinearity for (\widehat L^p)-initial data, Proc. Am. Math. Soc. 140, No. 11, 3905-3920 (2012). ZBL1283.35126.`:
Given $$\hat {L^p}:=\{ \phi: \hat{\phi} \in L^{p'} \}$$ if $p\geq2$
$$\hat{L^p} \subseteq L^p$$ Authors only said it is due to the Hausdorff-Young inequality but in my opinion this inequality can be used only for the proof of $$L^p \subseteq \hat{L^p}, \quad \text{if} \quad p<2$$
The two inequalities are equivalent by the Fourier inversion theorem. Indeed, if $f∈ L^p$ with $p≥ 2$, then $p'≤2$ and so defining $g = \mathcal F^{-1}(f)$ the inverse Fourier transform of $f$, by what you call Hausdorf-Young's inequality, it holds $$ \|f\|_{L^p} = \|\widehat g\|_{L^p} ≤ \|g\|_{L^{p'}} = \|\widehat f\|_{L^{p'}}. $$ which gives you your first inclusion.