The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form $$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q} \||x|^\beta u\|_{L^r},$$ for any $u \in C_c^\infty(\mathbb{R}^d)$ (there are constraints on all the parameters I won't mention here).

I am interested in whether similar such inequalities are known (or known to be false) for non-radial weights of a specified form (the encouragingly named paper Hardy and Caffarelli-Kohn-Nirenberg Inequalities with Nonradial Weights does not seem to answer this question; they consider a different class of weights). In particular, I am interested in the question of whether any inequalities of the following form hold. For $u \in C_c^\infty(\mathbb{R}^2)$, $$\||x|^\gamma u\|_{L^p} \leq C(\||x_2|^\alpha \partial_{x_1} u\|_{L^2} + \||x_1|^\alpha \partial_{x_2} u\|_{L^2})\||x|^\beta u\|_{L^r}.$$ I really only care about the case where $\alpha \in (0,1)$ and $\gamma \leq 0, \beta \geq 0$. This problem arises in trying to get estimates on certain degenerate parabolic equations.