$\mathscr{F}_{p_1}f=\mathscr{F}_{p_2}f$ $\nu$-almost everywhere on $i{\mathfrak{a}}^*$ as claimed.

2.  Since $\mathscr{I}_{q_j}h\in L^{q_j^\prime}({\mathfrak{a}},d\mu)^W$ for $j=1,2$, the claim follows from the injectivity of $\mathscr{F}_p$ for $p\in(1,2]$ and (i).

The remainder of the section is concerned with interpolation results in Lorentz spaces that will be needed in our proof of the Hardy–Littlewood inequality. The interested reader may consult [@Stein-Weiss-analysis Chapter V] for detailed proofs and historical remarks. Let $(X,\mu)$ be a $\sigma$-finite measure space and let $p\in(1,\infty)$. Define $$\|f\|^*_{p,q}=\begin{cases}\displaystyle \Bigl(\frac{q}{p}\int_0^\infty t^{q/p-1}f^*(t)^q\,dt\Bigr)^{1/q}&\text{if } q<\infty\\ 
\displaystyle \sup_{t>0} t\lambda_f(t)^{1/p}&\text{when } q=\infty\end{cases}$$ where $\lambda_f$ is the distribution function of $f$ and $f^*$ the non-increasing rearrangement of $f$, that is $$\lambda_f(s) =\mu(\{x\in X\,:\, |f(x)|>s\}) \quad\text{and}\quad f^*(t)=\inf \{s\,:\,\lambda_f(s)\leq t\}.$$ By definition, the Lorentz space $L^{p,q}(X)$ consists of measurable functions $f$ on $X$ for which $\|f\|^*_{p,q}<\infty$.

Let $(X,d\mu)$ and $(Y,d\overline{\nu})$ be $\sigma$-finite measure spaces. A linear operator $T:L^p(X,d\mu)\to L^q(Y,d\overline{\nu})$ is *strong type $(p,q)$* if it is continuous on $L^p(X,d\mu)$. Moreover, $T$ is *weak type $(p,q)$* if there exists a positive constant $K$ independent of $f$ such that for all $f\in L^p(X,d\mu)$ and all $t>0$, $$\mu\bigl(\bigl\{y\in Y\,:\, |Tf(y)|>t\bigr\}\bigr)\leq \Bigl(\frac{K}{s}\|f\|_{L^p(X,d\mu)}\Bigr)^q.$$ The infimum if such $K$ is the weak type $(p,q)$ norm of $T$.

For $1\leq p,q\leq\infty$, $L^{p,p}(X,d\mu)=L^p(X,d\mu)$, and $\|f\|^*_{p,q_2}\leq\|f\|^*_{p,q_1}$ whenever $q_1\leq q_2$. The 