Is the an example of function $f\in L^p(0,\infty)$ but $f\not \in L^q(0,\infty)$ given $q

Question

I been reading This, and I want to find some function on those case they are not inclusion each other. They give an example for $q<p$, $(x+1)^{-1/p}$ is a function in $L^q(0,\infty)$ but not in $L^p(0,\infty)$.

So I think there should be also some function $f\in L^p(0,\infty)$ but $f\not \in L^q(0,\infty)$ given $q<p$. But I failed to construct one.

score 2 · Accepted Answer · answered Nov 08 '18 at 07:25

2

Take $f(x)=x^{a}$ for $x>1$, $0$ for $x\leq 1$ where $-\frac 1 q < a <-\frac 1 p$.

answered Nov 08 '18 at 07:25

Kavi Rama Murthy

311,013

Is the an example of function $f\in L^p(0,\infty)$ but $f\not \in L^q(0,\infty)$ given $q

1 Answers1