Young's Inequality for Convolutions; when $r = infty$
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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 $1leq p, q leq infty$ , $frac{1}{p}+frac{1}{q}geq 1$ , and $frac{1}{r}=frac{1}{p}+frac{1}{q}-1$ . If $f in L^{p}, g in L^{q}$ , then $fast g in L^{r}$ and $||fast g||_{r} leq ||f||_{p}||g||_{q}$ . 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
