Prove that a functional on Sobolev space type is convex

Question

Let $a\in(-1,1)$, let: $$ W_a=\biggl\{ g\in W^{1,1}_{\text{loc}}(0,\infty): \int_0^\infty t^a|g(t)|^2\,dt<\infty,\text{ and } \int_0^\infty t^a|g'(t)|^2\,dt<\infty \biggr\},$$ let $\lambda>0$, i have to prove that the following functional: $$ G_\lambda(g)=\int_0^\infty t^a(\lambda|g(t)|^2+|g'(t)|^2)\,dt,\quad\forall g\in W_a,$$ is convex. So I think that I have to prove that for all $\alpha\in[0,1]$ and $f,g\in W_a$: $$ G_\lambda(\alpha g+(1-\alpha)f)\leq \alpha G_\lambda(g)+(1-\alpha)G_\lambda(f). $$ But I don't know how to go on. It seems to me more difficult than it seems. Any help would be appreciated.

Answer 1

My assert follow immediately by the convexity of the function $x\mapsto x^2$.