I bumped into the following convolution inequality mentioned without proof in a paper:
There is a constant $C > 0$ such that $$\int_{\mathbb{R}^n\times \mathbb{R}^n} K(x-y) u(x)u(y) \,d x\, d y \leq C \|K\|_{L^{p,\infty}(\mathbb{R}^n)} \|u\|^2_{L^q(\mathbb{R}^n)}$$ where $\dfrac{1}{q} = 1 - \dfrac{1}{2p}$ and $\|\cdot\|_{L^{p,\infty}(\mathbb{R}^n)}$ denotes the Marcinkiewicz "norm" on the weak $L^p$ space $$\|K\|_{L^{p,\infty}(\mathbb{R}^{n})} = \sup_{\lambda > 0} \lambda|\{x \in \mathbb{R}^n \colon |K(x)| > \lambda\}|^{1/p}.$$ Here $|A|$ is the Lebesgue measure of $A \subset \mathbb{R}^n.$
Does anyone know a reference to this result? The author claims that it is a well-known convolution inequality, but I do not know where to find it.
First you use Hölder's inequality for $u(y)$ and $(K*u)(y)=\int_{\mathbb{R}^n} K(y-x)u(x)\,dx$ with exponents $q$ and $q'$, respectively. Now use Young's inequality for weak-type spaces (Theorem 1.4.24, from the book of Grafakos that Jose quoted): if $1 <p,q, r <\infty $ are such that \begin{equation*} \frac{1}{r}+1=\frac{1}{p}+\frac{1}{q}, \end{equation*} then there is a positive constant $C = C(p, q, r)$ such that for any $f \in L^q(\mathbb{R}^n)$ and $g \in L^{p,\infty }(\mathbb{R}^n)$ \begin{equation*} \|f*g\|_{r}\leq C \|g\|_{p,\infty}\|f\|_{q} \end{equation*} (in your case, with $f=u$, $g=K$ and $ r = q'$). So
\begin{equation*} \frac{1}{p} + \frac {1} {q}=\frac{1}{q'} + 1=\left(1-\frac{1}{q}\right) + 1 , \end{equation*} which is the same as $2-\frac{1}{p}=\frac {2} {q}$ and the result follows.