# Thread: Convolution of Functions on Lp norm

1. ## Convolution of Functions on Lp norm

Let $\displaystyle 1\leq p\leq\infty$. Show that if $\displaystyle f\in L^p(\mathbb{R}^d)$ and $\displaystyle g\in L^1(\mathbb{R}^d)$, then $\displaystyle f\ast g\in L^p(\mathbb{R}^d)$ and also that:

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

It goes on to give a hint that says for $\displaystyle 1<p<\infty$ write:

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

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