Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive function on $\mathbb{R}_{+}$. Is it true that $$\left(\int_{0}^{\infty}(f^{*}(s))^{p'}(g(s))^{-p'}ds\right)^{p/p'}\leq c\int_{0}^{\infty}(f^{*}(s))^{p}(g(s))^{-p}s^{\frac{p}{p'}-1}ds$$ for some positive constant $c>0$ (later added: possibly depending on function $g$)?

Maybe it helps that, for $p, q\in (0, +\infty)$, the Lorentz space is defined via the quasi-norm $$||f||_{p, q}=\left(\int_{0}^{\infty}(f^{*}(s))^{q}s^{\frac{q}{p}-1}ds\right)^{1/q}$$ and we have the Hoelder inequality $$||fg||_{p, q}\leq C||f||_{p_{1},q_{1}}||g||_{p_{2},q_{2}},$$ where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.

Any help is appreciated!

Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive function on $\mathbb{R}_{+}$. Is it true that $$\left(\int_{0}^{\infty}(f^{*}(s))^{p'}(g(s))^{-p'}ds\right)^{p/p'}\leq c\int_{0}^{\infty}(f^{*}(s))^{p}(g(s))^{-p}s^{\frac{p}{p'}-1}ds$$ for some positive constant $c>0$ (later added: possibly depending on function $g$)?

Maybe it helps that, for $p, q\in (0, +\infty)$, the Lorentz space is defined via the quasi-norm $$||f||_{p, q}=\left(\int_{0}^{\infty}(f^{*}(s))^{q}s^{\frac{q}{p}-1}ds\right)^{1/q}$$ and we have the Hoelder inequality $$||fg||_{p, q}\leq C||f||_{p_{1},q_{1}}||g||_{p_{2},q_{2}},$$ where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.

Any help is appreciated!

In the case when $g$ is non-decreasing, see here.

Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive function on $\mathbb{R}_{+}$. Is it true that $$\left(\int_{0}^{\infty}(f^{*}(s))^{p'}(g(s))^{-p'}ds\right)^{p/p'}\leq c\int_{0}^{\infty}(f^{*}(s))^{p}(g(s))^{-p}s^{\frac{p}{p'}-1}ds$$ for some positive constant $c>0$?

Maybe it helps that, for $p, q\in (0, +\infty)$, the Lorentz space is defined via the quasi-norm $$||f||_{p, q}=\left(\int_{0}^{\infty}(f^{*}(s))^{q}s^{\frac{q}{p}-1}ds\right)^{1/q}$$ and we have the Hoelder inequality $$||fg||_{p, q}\leq C||f||_{p_{1},q_{1}}||g||_{p_{2},q_{2}},$$ where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.

Any help is appreciated!

Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive function on $\mathbb{R}_{+}$. Is it true that $$\left(\int_{0}^{\infty}(f^{*}(s))^{p'}(g(s))^{-p'}ds\right)^{p/p'}\leq c\int_{0}^{\infty}(f^{*}(s))^{p}(g(s))^{-p}s^{\frac{p}{p'}-1}ds$$ for some positive constant $c>0$ (later added: possibly depending on function $g$)?

Maybe it helps that, for $p, q\in (0, +\infty)$, the Lorentz space is defined via the quasi-norm $$||f||_{p, q}=\left(\int_{0}^{\infty}(f^{*}(s))^{q}s^{\frac{q}{p}-1}ds\right)^{1/q}$$ and we have the Hoelder inequality $$||fg||_{p, q}\leq C||f||_{p_{1},q_{1}}||g||_{p_{2},q_{2}},$$ where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.

Any help is appreciated!