Let $v(x,z)$ be a function, where $x\in\mathbb{R}^d$ is a $d$-dimensional real vector and $z\in(0,\infty)$. Let $D\subset \mathbb{R}^d$ be a domain and $|\alpha|<1$. Consider the space $L^2(z^{\alpha},D)$ with its norm defined by $$ \|v\|^2_{L^2(z^{\alpha},D)}=\int_{0}^{\infty}\int_{D}z^{\alpha}|v|^2dxdz. $$
I am trying to apply Holder's inequality, for example $\|fg\|_{L^2(D)}\leq \|f\|_{L^{\infty}(D)}\|g\|_{L^2(D)}$. In this weighted space above, I am wondering how should we do it. Does the following make sense? $$ \|vw\|_{L^2(z^{\alpha},D)}\leq \max_{z>0}\|w\|_{L^{\infty}(D)}\|v\|_{L^2(z^{\alpha},D)} $$