Norm of pointwise product of Lp functions

Question

Does the following inequality hold in $L_p$ spaces?

$\|fg\|_p\leq\|f\|_p\|g\|_p$

How would I go about proving this? Do I need to apply Cauchy Schwarz?

Apparently the solution link contains a counterexample. – Carucel Jan 14 '16 at 13:01 — Carucel, Jan 14 '16 at 13:01

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

1

Apparently this is not true, according to link:

If $\Omega = (0,1)$ and $f(x) = g(x) = x^{-1/3}$, we have $\|fg\|_2 = \infty$ but $\|f\|_2 = \|g\|_2 < \infty$.

edited Apr 13 '17 at 12:19

Community

answered Jan 14 '16 at 13:04

Carucel

1 Answers1