# Is there a weak version of Hölder's inequality?

If $f\in L_{w}^{p}$(weak $L^p$ space), $g\in L^{q}$, then does the inequality $$\|fg\|_{r}\leq C\|f\|_{p,w}\|g\|_q,\quad \text{where} \frac{1}{r}=\frac{1}{p}+\frac{1}{q}$$ hold ? Where $\|f\|_{p,w}=\sup_{t}(t^p\mu\{x||f|>t\})^{\frac1p}$.

Edit: The purpose I posed this question(which is false) is that I want to prove the following fact.If $f\in L_{w}^{p}$, $g\in L_{w}^{p'}$, $h\in L^{q}$, where $\frac{1}{p}+\frac{1}{q}<1$, then $$\|f(g\ast h)\|_{q}\leq C\|f\|_{p,w}\|g\|_{p',w}\|h\|_q.$$

Since $g\ast h\in L^r$ (Generalized Young), where $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$.

-

Let $(e,\infty)$ be the domain of $f$ and $g$, where $f(x)=x^{-\frac{1}{p}}$ and $g(x)=x^{-\frac{1}{q}}(\log x)^{-\frac{1}{r}}$. Then $\|f\|_{p,w}=1$, $\|g\|_q=(\frac{p}{q})^{\frac{1}{q}}$, but $\|fg\|_r=\infty$.