Inequality in $H^2,\ H^1$ and $L^\infty$

Question

I am trying to solve the following exercise:

If

$$\|f\|_{H^{2}(0, T)} + \|g\|_{H^{1}(0, T ; H^{1}(\mathbb{R}\setminus[1,2]))} + \|g\|_{L^{\infty}(0, T ; H^{1}(\mathbb{R}\setminus[1,2]))}\leq C, \quad C\in \mathbb{R}, $$ then $$ \|f\|_{L^{\infty}(0, T)} + \|g\|_{L^{\infty}(0, T\times \mathbb{R}\setminus[1,2])}\leq C_1, \quad C_1\in \mathbb{R}. $$ How can I prove this ?

Answer 1

This is again a consequence of the Sobolev embedding theorem (see Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations, Theorem 8.8): There is a constant $c>0$, depending only on $T$, such that $\|f\|_{L^\infty}\leq c \|f\|_{H^1}\leq c\|f\|_{H^2}$.