For $1\leq p<q\leq\infty$ what function is in $L^p$ but not in $L^q$

Question

Let $1\leq p<q\leq\infty$ and $-\infty\leq a<b\leq\infty$ be arbitrary. What specific functions satisfy $f\in L^p(a,b)$ and $f\notin L^q(a,b)$?

Answer 1

I suppose a function $\psi(x)$ is in $L^n(a,b)$ if $\int_a^b|\psi(x)|^ndx$ converges. I also assume that $-\infty<a<b<+\infty$ then i guess $$\psi(x)=\frac{1}{(x-a)^\alpha}$$ will do for all $\alpha\in(q^{-1}, p^{-1})$

Answer 2

$L^{q}(X)\subseteq L^{p}(X)$ for $1\leq p\leq q\leq\infty$ for finite measure space $X$, this is by Holder's inequality.

For infinite measure, say, $L^{1}(1,\infty)$ and $L^{2}(1,\infty)$, consider $1/x\in L^{2}(1,\infty)$ and $1/x\notin L^{1}(1,\infty)$.

On the other hand, say, $L^{1}(0,1)$ and $L^{2}(0,1)$, consider $1/x^{1/2}\in L^{1}(0,1)$ and $1/x^{1/2}\notin L^{2}(0,1)$.