Young's inequality for convolutions for functions of bounded support If $$f\in L^P(\mathbb{R}^d), g\in L^q(\mathbb{R^d}), \; \frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},$$ then Young's inequality for convolutions states
$$\|f*g\|_{L^r}\leq\|f\|_{L^p} \|g\|_{L^q}.$$
In particular, for $r=2, p=2, q=1, d=1$, we have
$$\int_{-\infty}^{\infty} |f*g(x)|^2 dx \leq \int_{-\infty}^{\infty} |g(x)|^2 dx \cdot \left(\int_{-\infty}^{\infty}|f(x)| dx \right)^2.$$
I was wondering if there is an analogous result when $f,g$ are supported on different subsets of $\mathbb{R}^d$. Specifically, in the case $f\in L^2([c,d]), g(x-y) \in L^2(x\in [a,b], y\in [c,d])$, I think the inequality should be
$$\int_{a}^{b} |f*g(x)|^2 dx \leq \sup_{y\in [c,d]} \int_{a}^{b} |g(x-y)|^2 dx \cdot \left(\int_{c}^{d}|f(y)| dy \right)^2.$$
Does this follow from Young's inequality?
 A: By the Schwarz inequality
\begin{align}
\left| \int_{c}^d g(x-y) f(y) dy \right|^2 & \leq \left( \int_{c}^d |g(x-y)|\cdot  |f(y)| dy \right)^2  \nonumber \\
&   \leq \left( \int_c^d |g(x-y)|^2 |f(y)| dy \right) \left(\int_c^d |f(y)|dy \right),
\end{align}
integrating over $[a,b]$ and using Fubini's theorem, we obtain
\begin{align}
\int_a^b \left| \int_{c}^d g(x-y) f(y) dy \right|^2 dx &\leq \int_a^b \left( \int_c^d |g(x-y)|^2 |f(y)| dy \right) \left(\int_c^d |f(y)|dy \right) dx \nonumber \\
& = \int_c^d |f(y)|dy \int_a^b \int_c^d |g(x-y)|^2 |f(y)|dy dx \nonumber \\
&  = \int_c^d |f(y)|dy  \int_c^d  |f(y)| \int_a^b |g(x-y)|^2 dx dy \nonumber \\
& \leq  \int_c^d |f(y)|dy  \int_c^d  |f(y)| \left( \sup_{y\in [c,d]} \int_a^b |g(x-y)|^2 dx \right) dy \nonumber \\
&= \sup_{y\in [c,d]} \int_a^b |g(x-y)|^2 dx  \left(\int_c^d |f(y)|dy \right)^2.
\end{align}
In the case integrals are over the whole space we have 
\begin{align}
\int_{\mathbb{R}^n} |g(x-y)|^2 dx &=  \int_{\mathbb{R}^n} |g(x)|^2 dx
\end{align}
 and we obtain the standard form of Young's inequality without having to take the supremum.
