In my notes I found the following result without proof or reference:
Let $(\mathcal{M},\mu)$ be a Borel measure space, with $\mu$ positive and $\sigma$-finite. Let $g,h : \mathcal{M} \mapsto \mathbb{R}^+$ be two measurable functions.
If $\| g \psi \|_{L^2} \leqslant c \| h \psi \| _{L^2}$ holds for some dense set of test functions $\psi$, then for all $s \in [0,1]$ :
$$\| g^s \psi \|_{L^2} \leqslant c^s \| h^s \psi \| _{L^2}.$$
Does anyone know if this result is true ? Do you have a reference for the proof ?
Is it a consequence of the Riesz-Thorin / Stein-Weiss type interpolation theorems ? If so can you please explain how (see e.g. https://www.math.ucla.edu/~tao/247a.1.06f/notes2.pdf)
Thank you for any help.
I found an article by Kato "A generalization of the Heinz inequality" 1961 that proves this in a more general setting.
https://projecteuclid.org/download/pdf_1/euclid.pja/1195523678