Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form

We obtain the inequalities of the form $${\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u(x){\mathrm{\mid }}^{2}h(u(x))\mathrm{d}x\le C{\int }_{\mathrm{\Omega }}{\left(\sqrt{\mathrm{\mid }Pu(x)\mathrm{\mid }\mathrm{\mid }{\mathcal{T}}_H(u(x))\mathrm{\mid }}\right)}^{2}h(u(x))\mathrm{d}x+\mathrm{\Theta },$$ where $\mathrm{\Omega }\subset {\mathbf{R}}^n$ is a bounded Lipschitz domain, $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1}(\mathrm{\Omega })$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\mathcal{T}}_H(\cdot )$ is certain transformation of the monotone $C^1$ function $H(\cdot )$, which is the primitive of the weight $h(\cdot )$, and $\mathrm{\Theta }$ is the boundary term which depends on boundary values of $u$ and $\mathrm{\nabla }u$, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g.~to some variants of the Douglas formulae.