What information is contained in the function $p\mapsto ||f||_p$?

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.