The properties of r.i. norms and the fact that the underlying measure space is nonatomic guarantee that the fundamental function is well defined. Moreover, one has $$\begin{aligned}
\label{E:fundamental-relation}
\varphi _{X}(t)\varphi _{X'}(t)=t, \quad t\in [0,\mu (R)).\end{aligned}$$

Let $X$ and $Y$ be r.i. spaces over $(0,\infty )$ and let $I\colon [0,\infty )\to [0,\infty )$ be a nondecreasing function. Then $$\begin{aligned}
\label{T:Lenka-unrestricted}
\left \|  \int _{t}^{\infty }\frac{f(s)}{I(s)}\,{\mathrm{d}}s\right \|  _{Y(0,\infty )}
\le C_{1} \|f\|_{X(0,\infty )}
\quad \mbox{for every }f\in \mathcal{M}_{+}(0,\infty )\end{aligned}$$ holds true with some positive constant $C_{1}$ if and only if $$\begin{aligned}
\label{T:Lenka-nonincreasing}
\left \|  \int _{t}^{\infty }\frac{g(s)}{I(s)}\,{\mathrm{d}}s\right \|  _{Y(0,\infty )}
\leq C_{2} \|g\|_{X(0,\infty )}
\quad
\mbox{for every nonincreasing }g\in \mathcal{M}_{+}(0,\infty )\end{aligned}$$ is valid with some positive constant $C_{2}$. This result originated as a consequence [@CPS Corollary 9.8] of a more general principle established in [@CPS Theorem 9.5] in connection with sharp higher-order Sobolev-type embeddings and its extension to unbounded intervals was given in [@P Theorem 1.10].

An important corollary of the Hardy–Littlewood inequality  is the fact that if $f$ is a nonincreasing function on $(0,\infty )$ and $X$ is an r.i. space over $(0,\infty )$, then in fact one has $$\begin{aligned}
\label{E:corollary-of-HL}
\|f\|_{X(0,\infty )}=\sup \left \{  \int _{0}^{\infty }g^{*}(t)f(t)\,
{\mathrm{d}}t: \|g\|_{X'(0,\infty )}\leq 