Functional analysis problem about $L_p$ spaces

Question

Prove that if $\phi(x) \notin L_{\infty}([0;1])$ then there exists $f(t) \in L_{2}([0;1])$ such that $f(t)*\phi(t) \notin L_{2}([0;1])$

Answer 1

Let $A_n=\{x: n \leq |\phi (x)|<n+1\}$ and $a_n =\frac 1 {n\sqrt {\lambda (A_n)}}$ if $\lambda (A_n) \neq 0$, $a_n=0$ if $\lambda (A_n)=0$ (where $\lambda$ is the Lebesgue measure). Let $f =\sum a_n I_{A_n}$. Then $f \in L^{2}$ and $f\phi \notin L^{2}$ provided $\lambda (A_n)>0$ for infintely many $n$ which is true if $\phi \notin L^{\infty}$.