I am studying integrals of the form $\int (\Delta \rho)^{\alpha} f^{\beta}$ where $0<\alpha < 1, \beta \geq 0$ and $\rho, f \in C_c^{\infty}(\mathbb{R}^n).$ My goal is to obtain lower bound on this quantity in the following form: $$||\nabla \rho||_{L^2} ||\nabla f||^{\beta}_{L^2} \leq C\int (\Delta \rho)^{\alpha} f^{\beta}$$
for some choice of the function $f.$ Is such a type of inequality possible?
No, far from it. First, probably the exponent $\alpha$ on the left-hand side is missing? Else, your inequality behaves badly with respect to scaling $\rho \mapsto \lambda \rho$. And I guess the right-hand side should be $\int |\Delta \rho|^\alpha |f|^\beta$? For negative $\Delta \rho$ and $f$, I am not even sure how you would define non-rational powers (while still making sense of ``$\le$'') and of course, the inequality fails whenever the right-hand side is negative.
But even then: Take some $f \not\equiv 0$ and $\rho_k(x) = \phi(|x|/k - 1)$ for some $\phi \in C^\infty(\mathbb R)$ with $\phi(s) = 1$ for $s \le 0$ and $\phi(s) = 0$ for $s \ge 1$. Then there is no $C > 0$ such that your inequality holds for all $k \in \mathbb N$.
Generally speaking, I would be very skeptical to estimate the gradient of $f$ by $f$. In fact, fixing $\rho$ and varying $f$ would provide another counter example.
