I'm having trouble with an estimate that would be helpful in information geometry.

The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a compact manifold without boundary with $Vol(X) = 1$. We also assume that $g$ satisfies $\int_X g^{10} dx =1.$ I want to obtain a positive lower bound on the following functional: $$DF(g) := \frac{ \| \nabla g \|_{L^{5/3}(X)}}{ \sqrt{1 - \int_X g^5 dx} }$$

This bound will depend on $X$, but I would ideally want it to be done in any dimension. In the 1-dimension case, this can be done, but it seems harder in higher dimensions.

One possible approach is to try to use the spectrum of the 5/3-Laplace operator to do this. To do this, first rewrite $DF(g)$ as follows:

$$ \frac{ \| \nabla g \|_{L^{5/3}(X)}}{ \sqrt{1 - \int_X g^5 dx} }=\frac{ \| \nabla g \|_{L^{5/3}(X)}}{ \sqrt{\int_X (1 - g)(1+g+g^2+g^3+g^4) dx} }$$

From here, use Holder's inequality and the $L^{10}$ estimate to get the following inequality:

$$ \frac{ \| \nabla g \|_{L^{5/3}(X)}}{ \sqrt{\int_X (1 - g)(1+g+g^2+g^3+g^4) dx} } \geq \frac{ \| \nabla g \|_{L^{5/3}(X)}} {C \sqrt{\| 1-g \|_{L^{5/3}(X)}}} = \frac{ \| \nabla (1- g) \|_{L^{5/3}(X)}} {C \sqrt{\| 1-g \|_{L^{5/3}(X)}}} $$

This resembles the Rayleigh quotient for the 5/3-Laplace operator. It's not exactly right, because of the square root. The reason to think that this might not be too bad of an issue is that $g$ is positive, which restricts the scale. However, it is not the case that $\int_X 1-g~ dx = 0$, which I think I need to get the technical details of this approach to work.

Does anyone have any advice on an interpolation inequality or something similar which would be able to help with this?

I'm having trouble with an estimate that would be helpful in information geometry.

The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a compact manifold without boundary with $Vol(X) = 1$. We also assume that $g$ satisfies $\int_X g^{10} dx =1.$ I want to obtain a positive lower bound on the following functional: $$DF(g) := \frac{ \| \nabla g \|_{L^{5/3}(X)}}{ 1 - \int_X g dx }$$

This bound will depend on $X$, but I would ideally want it to be done in any dimension. In the 1-dimension case, this can be done, and it feels like something that should follow from a Sobolev embedding inequality, but I'm not seeing how to do it.

Does anyone have any advice on an interpolation inequality or something similar which would be able to help with this?

Edit: An earlier version of this question had a related functional, but it turned out to not have the desired property. With extra calculations, I've been able to simple it to a form that seems more likely to be true.