Consider a sequence $f_n \to f$ convergent in $L^2(\mathbb{R}^N) \cap L^1(\mathbb{R})$. Prove that the respective Fourier transforms $$ \hat{f_n} \to \hat{f} $$ in $L^p(\mathbb{R}^N) \ \forall p > 2$. $$ $$ Resolution:
By hipothesis, $\Vert f_n - f \Vert_2 \to 0$, where $\Vert * \Vert_2$ denotes the norm in $L^2(\mathbb{R}^N)$.
Let $p > 2$. Denoting $ \Vert * \Vert_p $ by the norm in $L^p(\mathbb{R}^N)$,
$$ 0 \leq \Vert \hat{f_n} - \hat{f} \Vert_p^p = \int_{\mathbb{R^N}} \vert \hat{f_n} - \hat{f} \vert^p \\ = \int_{\mathbb{R^N}} \vert \hat{f_n} - \hat{f} \vert^2 \ \vert \hat{f_n} - \hat{f} \vert^{p-2} $$ I know that $$ \Vert \hat{f_n} - \hat{f} \Vert_2 = \Vert \widehat{f_n - f} \Vert_2 = \dfrac{1}{(2\pi)^N} \Vert f_n - f \Vert_2 \to 0 $$ when $ n \to + \infty$.
Do we have $ \Vert \hat{f_n} - \hat{f} \Vert_{p-2}^{p-2} < C$, for some constant $ C \geq 0 $ ? Why?
Fourier transform of an $L^{1}$ function $f$ is bounded by $\|f\|_1$. $f_n \to f$ in $L^{1}(\mathbb R^{n})$ implies that $\|f_n\|_1$ is a bounded sequence. So $\int |\hat {f_n}-\hat f|^{p} \leq (\sup_x |\hat {f_n}(x)-\hat f(x)|)^{p-2} \|f_n-f\|_2^{2} \to 0$.