Factorization of $L^{p}$ spaces

Question

Is the following theorem true and if so does it have a name?

Theorem: Let $p$, $q$, $r\in\left(0, \infty\right]$ with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$. Every $L^{q}$-function is the product of a $L^p$-function and a $L^{r}$-function.

I believe the statement is correct but I can't find the corresponding literature I need.

It's true, $\lvert f\rvert = \lvert f\rvert^{q/p}\cdot \lvert f\rvert^{q/r}$. I don't think it has a name. — Daniel Fischer, Aug 03 '15 at 14:37

score 1 · Answer 1 · answered Oct 20 '18 at 14:52

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Correctness of the statement follows from $$ \lvert f\rvert = \lvert f\rvert^{q/p}\cdot \lvert f\rvert^{q/r}. $$ Like Daniel Fischer, I do not know whether it has a name.

answered Oct 20 '18 at 14:52

Davide Giraudo

172,925

Is the result in more variables still true just because of this? For example, can we say that $L^q(\mathbb{T}^2)= L^p(\mathbb{T}^2)\cdot L^r(\mathbb{T}^2)$ – Mark_Hoffman Feb 18 '20 at 19:58
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I think that it would hold, since we only need Hölder's inequality. – Davide Giraudo Feb 18 '20 at 22:13

Factorization of $L^{p}$ spaces

1 Answers1