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When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$.

This inequality is a special (or rather limit) case of Lyapounov's inequality: let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in \ell_p \cap \ell_q $, then

$$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q}$$

I am looking for the corresponding inequality for Orlicz norms:

Question: Given Young functions $\phi_1, \phi_2$ and $\phi_3$, and $f \in \cap_i \ell_{\phi_i}$ is there a bound of the form $\|f\|_{\phi_2} \leq $ some expression with $\|f\|_{\phi_1}$ and $\|f\|_{\phi_3}$ (so that the "expression" tends to 0 if one of the norms tends to 0 and while the other remains bounded).

The limit case (where $\phi_1$ and $\phi_3$ are the 1 and $\infty$ norms) could be done as follows (under the assumption that $\frac{\phi(x)}{x}$ is increasing and does not take the value $+\infty$). It's easier to do it with the Luxemburg norm (which is the same as the original one up to a factor of 2). Let $f \in \ell_1 \cap \ell_\infty$, then $$ \| f\|_\phi = \inf \lbrace b >0 \mid \sum_n \phi( \tfrac{|f(n)|}{b}) \leq 1 \rbrace $$ Bound: $\sum \phi(|x|) \leq \sum |x| \tfrac{\phi(|x|)}{|x|} \leq \big( \sup_x \frac{\phi(|x|)}{|x|} \big) \sum |x|$. Assuming that $\phi(x)/x$ is increasing, this means that

$$ \sum_n \phi( \tfrac{|f(n)|}{b}) \leq \frac{\phi(\|f\|_\infty / b)}{\|f\|_\infty} \|f\|_1. $$

In particular, if $b = \dfrac{\|f\|_\infty}{ \phi^{-1} \big(\tfrac{\|f\|_\infty}{\|f\|_1} \big) }$, where $\phi^{-1}$ is a inverse (or semi-inverse if $\phi$ is not strictly increasing) of $\phi$, the sum is smaller than one. Hence $$ \|f\|_\phi \leq \dfrac{\|f\|_\infty}{ \phi^{-1} \big(\tfrac{\|f\|_\infty}{\|f\|_1} \big) } $$

Some further comments:

  • If it helps remove some technicalities, I would be fine various extra assumptions such as $\phi_i: [0,\infty[ \to [0,\infty[$ are strictly increasing.

  • The above inequalities are also true in the $L_p$-spaces (if you add the obvious assumptions like $f \in L_1 \cap L_\infty$). They can be proved using Hölder's inequality (although $\|f\|_q^q \leq \|f\|_1 \|f\|_\infty^{q-1}$ can be proved using a much more naive argument [similar to the one above])

  • I found Hölder inequalities for Orlicz norms (as well as some inequalities about convolutions), but I could not find the above "interpolation" inequalities.

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