I've been trying to prove what seems to be a little generalized version of the Young's Inequality for Convolutions. Here's the statement of the Theorem:
Let $1\leq p, q \leq \infty$, $\frac{1}{p}+\frac{1}{q}\geq 1$, and $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1$. If $f \in L^{p}, g \in L^{q}$, then $f\ast g \in L^{r}$ and $||f\ast g||_{r} \leq ||f||_{p}||g||_{q}$.
First we may assume that $f,g$ are non-negative, for we can replace $f,g$ with $|f|,|g|$ if necessary.
In the case where $p,q,r \leq \infty$, I used the Generalized Hölder's Inequality for three functions with $\frac{1}{r}+\frac{1}{p_1}+\frac{1}{p_2} = 1$, where $\frac{1}{p_1}=\frac{1}{p}-\frac{1}{r}$, and $\frac{1}{p_2}=\frac{1}{q}-\frac{1}{r}$, and got my desired result.
However, I'm stuck trying to prove the case when $r=\infty$; that is when $\frac{1}{p}+\frac{1}{q}=1$, $||f\ast g||_{\infty} \leq ||f||_{p}||g||_{q}$ . My guess is that I need to consider the case when either $p$ or $q$ in infinite (so that the other equals 1), and when neither are infinite.
Any help would be appreciated, thanks!
It's really just Hölder's inequality. Fix $x$ and write $f_x(y) = f(x-y)$. The translation invariance of the integral gives you $\|f_x\|_p = \|f\|_p$ for all $1 \le p \le \infty$ so that $$|f \ast g(x)| \le \int |f(x-y)| |g(y)| \, dy \le \|f_x\|_p \|g\|_q = \|f\|_p \|g\|_q.$$