Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ , and let $$u, v \in L^p(\Omega) \quad \forall p \in[1,\infty)$$ Find the most general assumption on $a(x)$ such that: $$\int_{\Omega} a(x)u^3v dx$$ is finite.
In practice I should find what is the most "general" $L^q(\Omega)$ in which $a(x)$ belongs. I hope it is clear.
Thanks
let $$\frac{3}{p}+\frac{1}{q} +\frac{1}{r} =1$$ then $$\int_{\Omega} a(x) v(x) u^3 (x) dx \leq \left(\int_{\Omega } |a(x)|^r dx\right)^{\frac{1}{r}}\left(\int_{\Omega } |u(x)|^p dx\right)^{\frac{3}{p}}\left(\int_{\Omega } |v(x)|^q dx\right)^{\frac{1}{q}}$$ Since for any $r>1$ we can find $p>3 , q>1 $ such that $$\frac{3}{p}+\frac{1}{q} +\frac{1}{r} =1$$ it is enough that $a\in L^r (\Omega )$ for some $r>1.$