An inequality for convolution using Hölder inequality

Question

Let $p , q , r$ be three real numbers in $[1 , + \infty]$ such that $\frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r}$. If $f \in L^p({\mathbb{R}}^n)$, $g \in L^q({\mathbb{R}}^n)$ and $r < \infty$, I have to prove that $$ {|(f*g)(x)|}^r \leq {\|f\|}_p^{r - p} {\|g\|}_q^{r - q} \int_{{\mathbb{R}}^n} {|f(y)|}^p {|g(x - y)|}^q dy $$ and my indication is to use Hölder inequality for three functions; my intuition sais that $f$ and $g$ are two of that functions but I have to obtain the third. Can you help me with this problem please? Thank you very much.

Answer 1

Let us define $$\alpha = \left(\frac{1}{p} - \frac{1}{r}\right)^{-1}, \quad \beta = \left(\frac{1}{q} - \frac{1}{r}\right)^{-1}.$$ Notice that they are such that $$ \frac{1}{\alpha} + \frac{1}{\beta} + \frac 1 r = 1. $$ We have: $$\begin{align} |(f*g)(x)| &\leq \int_{\mathbb R^n} |f(y)|\,|g(x-y)|\,dy \\ &= \int_{\mathbb R^n} \left(|f(y)|^{1-\frac{p}{r}}\right) \, \left(|g(x-y)|^{1 - \frac{q}{r}}\right)\, \left(|f(y)|^{p} \, |g(x-y)|^{q}\right)^{\frac{1}{r}}\,dy \\ &= \int_{\mathbb R^n} \left(|f(y)|^{\frac{p}{\alpha}}\right) \, \left(|g(x-y)|^{\frac{q}{\beta}}\right)\, \left(|f(y)|^{p} \, |g(x-y)|^{q}\right)^{\frac{1}{r}}\,dy \end{align}$$ Now apply apply Holder's inequality with exponents $\alpha, \beta, r$ to obtain: $$\begin{align} |(f*g)(x)| &\leq \|f\|_p^{\frac{p}{\alpha}} \, \|g\|_q^{\frac{q}{\beta}} \, \left(\int_{\mathbb R^n} |f(y)|^{p} \, |g(x-y)|^{q}\,dy\right)^{\frac{1}{r}}. \end{align}$$ from which the statement follows by raising both sides to the power $r$ and simplifying.