I am asking myself when a $L^p\to L^q$ multiplication operator is continuous. The following should be true:
Let $a:[0,1]\to\mathbb{C}$ be a measurable function. Let $T_a: L^p([0,1])\to L^q([0,1])$, with $p,q\in[1,\infty]$, $p, be the operator of pointwise multiplication by $a$, i.e., $(T_af)(x):=a(x)f(x)$. Then $T_a$ is continuous if and only if $a=0$ (almost everywhere).
Any suggestion for a rigorous proof?