following version of Hölder’s inequality for Lorentz spaces is well-known.

Let $0<p,q,r\leq\infty$, $0<s_1,s_2\leq\infty$. Then $$\|f\cdot g\|_{r,s}^* \leq C_{p,q,s_1,s_2}\|f\|^*_{p,s_1}\|g\|^*_{q,s_2}$$ where $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\frac{1}{s_1}+\frac{1}{s_2}=\frac{1}{s}$.

The dual of $L^{p,q}(X,d\mu)$ is the space $L^{p^\prime,q^\prime}(X,d\mu)$, where $\frac{1}{p}+\frac{1}{p^\prime}=1=\frac{1}{q}+\frac{1}{q^prime}$, and the dual of $L^{1,q}(X,d\mu)$ is $\{0\}$ when $1<q<\infty$. The following interpolation theorem is classical and can be found as Theorem 3.15 in [@Stein-Weiss-analysis Chapter V]. It subsumes the interpolation theorem of Marcinkiewicz, for example.

\[thm.lorentz-inter\] Suppose $T$ is a subadditive operator of (restricted) weak types $(r_j,p_j)$, $j=0,1$, with $r_0<r_1$ and $p_0\neq p_1$, then there exists a constant $B=B_\theta$ such that $\|T\|^*_{p,q}\leq B\|f\|^*_{r,q}$ for all $f$ belonging to the domain of $T$ and to $L^{r,q}$, where $1\leq q\leq\infty$, $$\label{eqn.interpolate-indices}
\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1},\quad \frac{1}{r}=\frac{1-\theta}{r_0}+\frac{\theta}{r_1}\quad\text{and}\quad 0<\theta<1.$$

\[cor.HY-Lorentz\] If $f\in L^p({\mathbb{R}}^n)$, $1<p\leq 2$, then its Fourier transform ${\hat{f}}$ belongs to $L^{p',p}({\mathbb{R}}^n)$ and there exists a constant $B=B_p$ independent of $f$ such that $\|{\hat{f}}\|_{p',p}^*\leq B_p\|f\|_p$, where $\frac{1}{p}+\frac{1}{p'}=1$. In particular the Fourier transform is a continuous linear mapping from $L^p({\mathbb{R}}^n)$ to the Lorentz space $L^{p',p}({\mathbb{R}}^n)$ for $1<p<2$.

Taking $(r_0,p_0)=(1,\infty)$, $(r_1,p_1)=(2,2)$ in theorem \[thm.lorentz-inter\], the conditions in translate into $\frac{1}{p}=\frac{\theta}{2}$ and $\frac{1}{r}=1-\frac{\theta}{2}$, that is, $r=p'$. Furthermore take $q=r$. Since $\theta\in(0,1)$ in the hypothesis of theorem \[eqn.interpolate-indices\], the role of $p$ and $p'$ must be exchanged when 