# A convolution inequality for Marcinkiewicz spaces

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.

• If I recall correctly you can find this in "classical Fourier Analysis" by Grafakos. May 15, 2021 at 6:37

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 $$1 are such that $$\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.