I am trying to figure out an analysis problem related to Fourier transform and Young's convolution inequality. Here is the problem statement:
Let $f \in L^1(\mathbb{R}^n) \cap L^p(\mathbb{R}^n) \cap C^{\infty}(\mathbb{R}^n)$ for some $1<p<\infty$. Suppose the support of $\hat{f}$ (the Fourier transform of $f$) is contained in $B(0,R)$, ball centered at zero with radius $R$. For all $p \leq r < \infty$, we have $$\lVert D^{\alpha} f \rVert_{L^r} \leq CR^{\lvert \alpha \rvert + n(1/p-1/r)} \lVert f \rVert_{L^p} $$ (Hint: Write $D^{\alpha} f $ as a convolution and apply Young's convolution inequality.)
Young's inequality: Suppose $1 \leq p,q,r\leq \infty$ and $1+1/r=1/p+1/q$. Suppose $f \in L^p(\mathbb{R}^n)$ and $g \in L^q(\mathbb{R}^n)$, then $f * g \in L^r(\mathbb{R}^n)$ and $$ \lVert f*g \rVert_{L^r(\mathbb{R}^n)} \leq \lVert f \rVert_{L^p(\mathbb{R}^n)} \lVert g \rVert_{L^q(\mathbb{R}^n)} $$
I am currently stuck on the first step in the hint, which is to write $D^{\alpha}f$ as a convolution.
What I have done so far is to write $D^{\alpha}f$ as $$ D^{\alpha}f = \check{\widehat{D^{\alpha}f}} = \check{\widehat{D^{\alpha}f} \cdot \chi_{B(0,R)}} $$ where the two tiny "check" marks ("\widecheck" doesn't work...) is the inverse Fourier Transform.
Some hint on this would be nice!
Hint. $\hat f = \hat f \cdot \phi$, where $\phi$ is a smooth bump function satisfying $1_{B(0,R)} \le \phi \le 1_{B(0,2R)}$.
With the hint, and the property that $(f\cdot g)\check{} = \check f\ast \check g$, \begin{align*} f = \check{\hat f} &= (\hat f\cdot\phi)\check{} = \check{\hat f}\ast \check\phi = f\ast\check \phi. \end{align*} Because $\phi$ is a smooth bump function, $\check \phi$ is Schwartz, which should be good enough to plug into Young's inequality.
(Notation. For a set $E$, $1_E$ is the indicator function of $E$, equal to $1$ on $E$ and equal to $0$ on $E^c$.)