Background: Let $\Omega$ be a bounded domain, $f,g: \Omega \to \mathbb{R}$, with $f \in L^{1}(\Omega)$, $g \in L^{\infty}(\Omega)$. By Holder inequality, $fg \in L^1(\Omega)$ and $$ \| fg \|_{L^1(\Omega)} \leq \| f \|_{L^{1}(\Omega} \| g \|_{L^{\infty}(\Omega)}.$$ with $\|g\|_{L^{\infty}(\Omega)} = \operatorname{ess} \sup |g|$.
In [Sobolev Spaces, Adams] page 27, theorem 2.12 (Reverse Holder Inequality), $$ \| fg \|_{L^1(\Omega)} \geq \| f \|_{L^p(\Omega)} \| g \|_{L^q(\Omega)}$$ with $0 < p < 1$ and $p^{-1} + q^{-1} = 1$.
Question 1: Can I conclude in the limit $p \to 1^{-}$ and $ q \to -\infty$ that,
$$ \| fg \|_{L^1(\Omega)} \geq \| f \|_{L^1(\Omega)} (\operatorname{ess} \inf g) $$ holds?
Question 2: Does the following
$$ \int_{\Omega} |f| g ~\mathrm{d} x \geq \| f \|_{L^1(\Omega)} (\operatorname{ess} \inf g) $$
also hold? Intuitively it should make sense since the set $\{ x \in \Omega ~|~ g(x) < \operatorname{ess} \inf g \}$ is Lebesgue measure zero. But I want to be careful if I am missing any extra condition.