I have a question about a particular commutator estimate as it occurs in the study of Fokker-Planck equations with low regularity data, see e.g. (1,2).

Denote by $rho_varepsilon$ some usual regularizing kernel (mollifier) family and let $1 < r < 2$ be given and fixed. We have functions $$sigma in W^{1,infty}_{loc}(mathbb{R}^N)^{N times K} quadtext{and}quad p in L^infty(0,T;L^r(mathbb{R}^N)) quad text{with} quad sigma^top nabla p in L^2(0,T;L^r(mathbb{R}^N))$$ at hand. (No info on $nabla p$ itself.) Let $p_varepsilon := rho_varepsilon star p$ be the regularized $p$.

Define the mollification-commutator $(rho_varepsilon,D)(f) := rho_varepsilon star (Df) – D(rho_varepsilon star f)$ for some function $f$ and a differential operator $D$. Let $gamma in C^2(mathbb{R})$ with bounded first and second derivative and let $varphi$ be a test function on $(0,T) times mathbb{R}^N$.

I want to show that $$int_0^Tint_{mathbb{R}^N} varphi , gamma'(p_varepsilon) , r_varepsilon longrightarrow 0 quad text{as}~varepsilon to 0,tag{1}$$

where

$$r_varepsilon := rho_varepsilon star partial_i(sigma_{ik}sigma_{jk}partial_j p) – partial_i(sigma_{ik}sigma_{jk}partial_j p_varepsilon) = partial_i bigl((rho_varepsilon,sigma_{ik}sigma_{jk}partial_j)(p)bigr). $$

In (1,2) the case $r=2$ is considered where this indeed works out. (One also has $p in L^infty(0,T;L^infty(mathbb{R}^N)$ there, in addition to the integrability of $p$ introduced above, but this does not necessarily enter this argument under the assumptions above, as far as I see it.) They rewrite $r_varepsilon$ further into begin{align*}r_varepsilon &= partial_i bigl(sigma_{ik}(rho_varepsilon,sigma_{jk}partial_j)(p)bigr) + (rho_varepsilon,partial_isigma_{ik})(sigma_{jk}partial_j p) + (rho_varepsilon,sigma_{ik}partial_i)(sigma_{jk}partial_j p) \ &:= partial_i (sigma_{ik}R_varepsilon) + S_varepsilon + T_varepsilon.end{align*}

These commutators are then shown to converge to zero suitably by the commutator lemma (see below) which transfers to (1) by dominated convergence. However I cannot seem to make the one term with a derivative on the commutator $partial_i (sigma_{ik}R_varepsilon)$ work out in this setting since one needs to integrate by parts for this term in (1), and the resulting integrable bounds in the critical term $$int_0^T int_{mathbb{R}^N} varphi , gamma”(p_varepsilon) , sigma^top nabla p_varepsilon cdot R_varepsilon$$ are only $1/r + 1/r$-integrable in space (for $sigma^top nabla p_varepsilon$ and $R_varepsilon$), which works out to $1$ exactly for $r=2$.

I have tried modifying the splitting, but to no avail. I am willing to assume more or less any regularity on $sigma$. However I would like to avoid $sigmasigma^top$ being uniformly positive definite. (This would lead to $p(t) in W^{1,r}(mathbb{R}^N)$ and then I think I can do it, but this is the one assumption I would like to avoid.)

I had suspected (hoped) that with a second derivative on $sigma$ one could compensate lack of differentiability of $p$, but the proof of the commutator lemma seems to use a quite nice cancellation property which works only for first order combinations, so this led nowhere.

The motivation is to consider an inhomogeneous Fokker-Planck equation with data comparable to the cited works (1,2).

Any hints would be welcome.

**Lemma** (*Commutator lemma*, (3, Lemma II.1)). Let $1/beta = 1/alpha + 1/s$ and

- $g in L^{alpha}_{loc}(mathbb{R}^N)$ and $f_1 in L^s_{loc}(mathbb{R}^N)$, and
- $c in W^{1,alpha}_{loc}(mathbb{R}^N)$ and $f_2 in L^s_{loc}(mathbb{R}^N)$.

Then $$(rho_varepsilon,g)(f_1) to 0 quad text{and} quad (rho_varepsilon,c partial_i)(f_2) to 0, quadtext{each in}~L^beta_{loc}(mathbb{R}^N).$$

(1) *Le Bris, C.; Lions, P.-L.*, **Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients**, Commun. Partial Differ. Equations 33, No. 7, 1272-1317 (2008). ZBL1157.35301.

(2) *Luo, De Jun*, **Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients**, Acta Math. Sin., Engl. Ser. 29, No. 2, 303-314 (2013). ZBL1318.35130.

(3) *DiPerna, R. J.; Lions, P. L.*, **Ordinary differential equations, transport theory and Sobolev spaces**, Invent. Math. 98, No. 3, 511-547 (1989). ZBL0696.34049.