I am on a closed and bounded surface $S$ without boundary. Then it holds that $$ \int_S \nabla f \cdot \nabla f \mathrm dA = -\int_S f\Delta f \mathrm dA. $$ This is the $H^1(S)$ seminorm, meaning that I can express the $H^1(S)$ norm as $$ \|f\|_1 = \int_S f(1-\Delta) f \mathrm dA. $$
Imagine now I am on a weighted $L^2$ space with weight function $w$.
What can be said for $ \|f\|_1^w $? The problem becomes to integrate $$ \int_S w\nabla f \cdot \nabla f \mathrm dA. $$ I tried to play around with various integration by parts, but I end up getting gradients of the weights and so on. I see no reason why it shouldn't hold that $$ \|f\|_1^w = \int_S wf(1-\Delta)f \mathrm dA. $$ Here $\mathrm d A$ is just the Lebesgue measure, so if one says that we are instead integrating against $\mathrm dW = w\mathrm d A$, then it just seems like it would hold, but I don't know. Is it at all possible to rewrite in terms of only Laplacians? $$ \int_S w \nabla f \cdot \nabla f \mathrm d A? $$