Interpolation inequality related to the 5/3-Laplace operator

Question

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.

Answer 1

We have $$ DF(g)=\frac{\|\nabla g \|_{L^{5/3}(X)}}{ \|g\|_{L^{10}(X)} -\|g\|_{L^1(X)}},$$

which is a homogeneous expression. Hence we can drop the condition $\int_X g^{10}dx=1$ and replace it by $\|g\|_{L^1(X)}=\int_X g dx=1$. If $dim(X)=2$ we have by the Sobolev and the triangle inequality $$ DF(g)=\frac{\|\nabla g \|_{L^{5/3}(X)}}{ \|g\|_{L^{10}(X)} -\|g\|_{L^1(X)}}\geq \frac{C\|g-1 \|_{L^{10}(X)}}{ \|g\|_{L^{10}(X)} -1}\geq C. $$

For $dim(X)>2$ the Sobolev space $W^{1,5/3}(X)$ can not be embedded into $L^{10}(X)$, i.e we could find $g\in W^{1,5/3}(X)$ with $g \notin L^{10}(X)$ or with $\|g\|_{L^{10}(X)}$ arbitrarily large. Consider for example for $d\geq 3$ that $B_0(1)(\mathbb{R}^d) \subset X$ and $g(x)=|x|^{-d/10+\epsilon}$ in $B_0(1)(\mathbb{R}^d)$.