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It is well-known that if $1\leq p\leq q\leq \infty $ then

$$ L^p(X)\cap L^q(X)\subset L^r(X)\quad\quad \text{whenever $r\in [p,q]$}\tag{I}\label{Eq}.$$

Indeed let $u\in L^p(X)\cap L^q(X)$. For some $0\leq \theta\leq 1$ we can write $$\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q} $$

so that, $|u|^{1-\theta}\in L^{p/1-\theta}$ and $|u|^{\theta}\in L^{q/\theta}$ and the Hölder's inequality implies,

$$\|u\|_{L^r(X)}\leq \|u^{1-\theta}\|_{L^{p/1-\theta}(X)}\|u^{\theta}\|_{L^{q/\theta}(X)}= \|u\|_{L^{p}(X)}^{1-\theta}\|u\|_{L^{q}(X)}^{\theta}.$$

Replacing $\theta=\frac{\frac{1}{r}- \frac{1}{p}}{\frac{1}{q}- \frac{1}{p}}$ by its value gives

$$\|u\|_{L^r(X)}\leq \|u\|_{L^{p}(X)}^{\frac{\frac{1}{q}- \frac{1}{r}}{\frac{1}{q}- \frac{1}{p}}}\|u\|_{L^{q}(X)}^{\frac{\frac{1}{r}- \frac{1}{p}}{\frac{1}{q}- \frac{1}{p}}}$$

Question: What is the analogue of the property \eqref{Eq} for Orlicz spaces of type $L^\phi(\Bbb R^d)$? Any hint or references is welcome...

Recall the Orlicz space $L^\phi(\Bbb R^d)$: \begin{align*} L^\phi(\Bbb R^d)&= \Big\{u: \Bbb R^d\to \Bbb R\text{ meas.}:~ \int_{\Bbb R^d} \phi\Big(\frac{|u(x)|}{\lambda}\Big)d x<\infty ~~\text{for some $\lambda>0$}\Big\}. \end{align*} The space $L^\phi(\Bbb R^d)$ is equipped with the Luxemburg norm $\|\cdot\|_{L^\phi(\Bbb R^d)}$ by \begin{align} \|u\|_{L^\phi(\Bbb R^d)}=\inf \Big\{ \lambda>0~: \int_{\Bbb R^d} \phi\Big(\frac{|u(x)|}{\lambda}\Big)d x\leq 1\Big\}. \end{align}

Where $\phi$ is a sufficiently nice Young function, e.g., $\phi$ is continuous, increasing, convex and in addition, the mapping $x\mapsto \frac{\phi(x)}{x}$, $x>0$ is increasing and satisfies \begin{align*} &\lim_{x\to 0^+}\frac{\phi(x)}{x}= \lim_{x\to \infty}\frac{x}{\phi(x)}= 0. \end{align*}

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    $\begingroup$ Well, $u^r \leq \max( u^p, u^q)$ for every $u\in [0,\infty)$, and this gives another proof that $L^p\cap L^q \subset L^r$. So the obvious modification using Young's functions should still work. $\endgroup$ Apr 16 '21 at 2:50
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    $\begingroup$ By the way, I think you should be asking about interpolation spaces and not just the intersection space. There are a lot of spaces that contain $L^\phi \cap L^\psi$. The interpolation spaces have the property that the norm can be bounded by something like $\|u\|_{L^\phi}^\theta \|u\|_{L^\psi}^{1-\theta}$. In that vein, Googling finds me eudml.org/doc/218150 $\endgroup$ Apr 16 '21 at 3:00
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It's many years ago that I read it, but I think that some of the most general interpolation type results for Orlicz spaces were contained in O’Neil, Richard, Integral transforms and tensor products on Orlicz spaces and $L(p,q)$ spaces, J. Anal. Math. 21, 4–276 (1968) which is practically a monograph. Some other interpolation results are in the monograph Rao, M. M. and Ren, Z. D., Theory of Orlicz spaces, Marcel Dekker, New York, 1991.

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