# Inequality in Lp spaces with 0<p<1

I just asked if the following inequality is correct ?

${\parallel f\parallel}_{{L}_{p}} \leq c {\parallel f\parallel}_{{L}_{2}}$

Where ${L}^{p}$ is the space of periodic real functions on a bounded set $Q$ ,such that $Q$$\subset$ ${R}^{3}$ with $0<p<1$ .

I am loking for a reference to the ${L}^{p}$ spaces with $0<p<1$ ? . Thanks.

• The statement is not clear. The integrals in the norm taken are supposed to be over Q since the integrals over ${R}^{3}$ are not finite, Where does periodicity come in? Anyway, the inequality follows easily from Holder's inequality when you take the integrals over Q. – Kavi Rama Murthy Nov 3 '17 at 9:32
• $L^{p}$ spaces with $0<p<1$ are not found in basic texts because they are not normed vector spaces. They are badly behaved topological vector spaces even though they are metrizable, They have trivial dual spaces. – Kavi Rama Murthy Nov 3 '17 at 9:36
• Here we consider the $Lp$ space of periodic functions i.e$\dot{L}p$ . In this space we can use the Fourier transform. Since $p<1$ we get $1/p$>$1$ , then ,I can't see how to use Holder's inequality . – Kato Nov 3 '17 at 10:25