Approximation of Sobolev function by smooth functions with smaller integral norms

Question

Suppose $f\in L_1^2(B)$, $B\subset \mathbb R^4$. Can we find a smooth sequence such that $$ f_i\to f\in L_1^2\quad\&\quad \int_B|df_i|^2\leq\int_B|df|^2 \quad\&\quad \int_{B}|f_i|^4\leq\int_B|f|^4? $$ I only know that $df_i\to df$ in $L^2(B)$ How to get the subsequence such that the middle one holds? (I suppose we know the fact that $C^\infty$ is dense in $L_1^2(B)$) By Sobolev embedding, $L_1^2\to L^4$ is not compact, that seems there are counterexamples that $f_i\to f$ in $L_1^2$ but $\int_B|f_i|^4\to\infty$?

Answer 1

Take any sequence of smooth functions $g_i$ such that $g_i\to f$ in $L^2_1$. By definition of the Sobolev norm, $dg_i\to df$. By the Sobolev embedding, $g_i\to f$ in $L^4$. In particular, $$ \int_B|dg_i|^2 \to \int_B|df|^2 \quad \text{ and } \quad \int_{B}|g_i|^4 \to \int_B|f|^4 \tag1 $$ Let $$r_i = \max \left( \left(\frac{\int_B|dg_i|^2}{\int_B|df|^2}\right)^{1/2}, \left(\frac{\int_{B}|g_i|^4 }{\int_B|f|^4}\right)^{1/4} \right) $$ By $(1)$ we have $r_i\to 1$. Therefore the functions $f_i=r_i^{-1}g_i$ converge to $f$ in $L^2_1$ norm (using $\|f_i-g_i\| = |1-r_i^{-1}|\|g_i\|\to 0$). They satisfy the required inequalities by construction.

---

It's also possible to prove that mollification of $f$ directly produces the functions that obey those inequalities, but that looks like work. Work is bad.