$\text{supp}(f) = \text{supp} (g) , \|f\|_{L^{2}(\mathbb R)} \leq \|g\|_{L^{2}(\mathbb R)} \implies \|f\|_{L^{2}([0,1])} \leq \|g\|_{L^{2}([0,1])}$?

Question

Let $f,g \in L^{2}(\mathbb R)$. Suppose that $\int_{\mathbb R} |f(t)|^{2} dt \leq \int_{\mathbb R} |g(t)|^{2} dt.$

We also assume that support of $f$ and support of $g$ are equal.

My Question is: Can we say $\int_{0}^{1} |f(t)|^{2} dt \leq \int_{0}^{1} |g(t)|^{2} dt$ ?

Answer 1

No, certainly not. Just consider $f=2\cdot 1_{[0,1]}+1_{[2,10]}$ and $g=1_{[0,1]}+1000\cdot 1_{[2,10]}$.