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 $\|f*g\|_p\leq\|f\|_p\|g\|_1$ .

Author: mathtuition88

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