Young's inequality for convolutions for functions of bounded support

Question

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?