What is the relationship between $p$-norm and 2-norm for any $p$-norm?

Question

I know that there are some general inequalities between the $2$-norm and $1$-norm or $\infty$-norm

Suppose I am given an arbitrary $p$-norm, obviously $p \geq 1$

What can we say about the inequality between $\|x\|_2$ and $\|x\|_p$?

Answer 1

The answer depends on the underlying measure space. The inequality $\|x\|_p\leq\|x\|_r$ does not always hold for $p>r$. For instance, if $X=(0,1)$ with the Lebesgue measure, then it is not true that $\|f\|_2\leq \|f\|_1$. For example, the function $x^{-1/2}$ belongs to $L_1(0,1)$ but not to $L_2(0,1)$. In fact, if $(X,\Sigma,\mu)$ is a probability space, i.e., $\mu(X)=1$, then then $\|f\|_r\leq \|f\|_p$ if $r<p$, which is the reverse inequality of that suggested in the first comment above.