Normalize a function and a measure so that the $L^p$ norm is $1$ for two values of $p$

Question

I'm reading Tao's book on the interpolation of $l^p $ spaces and one part writes "if $\|f\| _{ L^{p _ 0} } = \|f \|_ {L ^{ p _ 1 }} =1 $ then we are done. To obtain the general case, one can multiply the function $f$ and the measure $\mu$ by appropriately chosen constants to obtain the above normalization."

I have no idea how can I get this normalization since I feel if I'm given a functions $f$ and a measure $\mu$, no matter how I normalize them, $\|f\| _{L^{ p _ 0 }} \neq \|f \|_{L^{ p _ 1 }}$ in general.

Answer 1

• Multiplying $f$ by a constant $a>0$ results in all norms multiplied by $a$.
• Multiplying $\mu$ by a constant $b>0$ results in $L^p(\mu)$ norm multiplied by $|b|^{1/p}$.

So, you want $$ab^{1/p_0} = 1/\|f\|_{p_0},\qquad ab^{1/p_1} = 1/\|f\|_{p_1}$$ and this system can be solved for $a$ and $b$.