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.

  • $\begingroup$ 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. $\endgroup$ – Kavi Rama Murthy Nov 3 '17 at 9:32
  • $\begingroup$ $L^{p}$ 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. $\endgroup$ – Kavi Rama Murthy Nov 3 '17 at 9:36
  • $\begingroup$ 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 . $\endgroup$ – Kato Nov 3 '17 at 10:25

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