Given a measurable function $f:\mathbb{R}\rightarrow\mathbb{R}$, we obtain a function $\nu_f:(0,\infty)\rightarrow [0,\infty]$ defined by $\nu_f(p):=||f||_p$

This function $\nu_f$ will not necessarily be finite almost everywhere, but it should still contain a lot of information about $f$.

$\nu_f$ will be a continuous function wherever it is finite. Will $\nu_f$ satisfy any stronger types of continuity, or have other nice properties? When will it be differentiable, and what interpretation does the derivative $\nu_f'$ have at a fixed value of $p$?

Is there a name for the function $\nu_f$ or the mapping $f\mapsto\nu_f$? What other information can we deduce about $f$? What can we say about two functions $f,g$ whose associated functions $\nu_f,\nu_g$ are equal?

Sorry about all the questions, but I'm positively thirsting for information!


The thirst can be alleviated by reading Tao's post Interpolation of $L^p$ spaces.

A fundamental property of $\nu_f$ is log-convexity: $\log \nu_f$ is a convex function of $1/p$. Tao gives four proofs of this fact. The fourth proof exhibits another nice property: extending the range of $p$ to a vertical strip in complex plane (whose base is an interval $p_0<p<p_1$ in which the norm is finite) we get a holomorphic function of $p$. Hence, $\nu_f$ is infinitely smooth in the interior of the domain, and the derivative of $\nu_f^p$ is given by differentiation of $\int |f|^p$, which gives an integral with extra $\log|f|$ in it. This kind of integral represents Shannon entropy, and differentiation with respect to $p$ can be used to prove some inequalities for said entropy.

Tao also remarks that $\nu_f $ is given by the moments of the distribution function $\lambda_f(t)=\mu\{|f|> t\}$ via
$$\nu_f^p = p \int_0^\infty t^{p } \lambda_f(t)\,\frac{dt}{t} \tag1 $$

Thus, the most information we could hope to get out of $\nu_f$ is the distribution function of $f$. We cannot always do that: for example, if $\nu_f$ is finite at one point only, there is not much information to be obtained from it. I'll make a strong assumption: $f$ is in $ L^\infty$ and its essential support has finite measure. Say, $|f|\le M$ a.e. The integral (1) goes from $0$ to $M$ only. Write $t=Me^{-x}$, obtaining $$\nu_f^p = p M^p\int_0^\infty e^{-px} \lambda_f(Me^{-x})\,dx \tag2 $$ This is the Laplace transform of $x\mapsto \lambda_f(Me^{-x})$. The latter function is bounded (because of the support having finite measure), hence it can be recovered from the Laplace transform by the inverse transform.

The conditions can surely be weakened somewhat, but I'd need to read up on the invertibility of the Mellin transform... since you're thirsty for knowledge, I leave that to you.


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