## Young’s Convolution Theorem

Let $1\leq p,q\leq \infty$ and $1/p+1/q\geq 1$, and let $1/r=1/p+1/q-1$. If $f\in L^p(\mathbb{R}^n)$ and $g\in L^q(\mathbb{R}^n)$, then $f*g\in L^r(\mathbb{R}^n)$ and $\displaystyle \|f*g\|_r\leq\|f\|_p\|g\|_q.$

Amazing Theorem! If $q=1$, then $\|f*g\|_p\leq\|f\|_p\|g\|_1$.

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