# Convolution of Functions on Lp norm

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• Sep 23rd 2010, 08:17 PM
mathematicalbagpiper
Convolution of Functions on Lp norm
Let $1\leq p\leq\infty$. Show that if $f\in L^p(\mathbb{R}^d)$ and $g\in L^1(\mathbb{R}^d)$, then $f\ast g\in L^p(\mathbb{R}^d)$ and also that:

$||f\ast g||_{L^p(\mathbb{R}^d)}\leq ||f||_{L^p(\mathbb{R}^d)}||g||_{L^1(\mathbb{R}^d)}$.

It goes on to give a hint that says for $1 write:

$|(f\ast g)(x)|\leq\int_{\mathbb{R}^d}[|f(y)|\cdot|g(x-y)|^{1/p}]|g(x-y)|^{1/q}dy$.

where $q$ is the conjugate exponent of $p$, and apply Holder's Inequality.