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Dear fellow mathematicians,

The Markov inequality reads, for $(\Omega, \mathcal{F}, \mu)$ being a measure space, and $f$ a real valued function on $\Omega$ (you can also see Stein, Singular Integrals and Differentiability Property of functions, p4-5):

$\forall \epsilon > 0, \mu\{\omega \in \Omega: |f(\omega)| > \epsilon \} \leq \frac{1}{\epsilon} \|f\|_{L^1(\Omega,\mu)}$

I was quite happy with that so far. Unfortunately, I am now dealing with some more complicated stuff... at least more complicated to me!

My question is, what happens to that inequality, if we no longer consider functions $f:\Omega \mapsto \mathbb{R}$ but functional $F:L^p(\Omega,\mu) \mapsto \mathbb{R}$?

My first problem is what about the measure used on left-hand side of the inequality? Does such an inequality makes sense? Or could it make sense? (I was thinking for instance of how 'dense' are the functions having an $L^p$ norm smaller than a given threshold? )

If you don't know the answer, you might know the keywords I should have a look at? Any tip is welcomed!

Thanks in advance for your kind help.

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