Chain rule for weakly differentiable functions

Question

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap L^1(\mathbb R^n,fdx)$. Is it true that $$ f\nabla \log f = \nabla f, $$ again in the sense of distributions?

Obviously the result is true if $f\in C^1(\mathbb R^n)$; it would also be true if $\log f $ were a Sobolev function with bounded range, since then the mapping $\log f \mapsto f$ (i.e. the exponential function $s\mapsto e^s$) can be considered $C^1$ and Lipschitz, and usual theorems on composition apply (e.g. Brezis, Functional Analysis, Sobolev Spaces, and PDEs, Corollary 8.11). In homogeneous space, i.e. without the weight in the $L^1$-space, the result would follow from regularization by convolution.

Without these helping properties, does anyone know how to prove this?

Answer 1

I think it works by cutting-off as follows: Fix $\epsilon>0$, let $f_\epsilon=\min\{1/\epsilon,\max\{\epsilon,f\}\}$, and observe that $f_\epsilon\to f$ in $L^1_{loc}$.

Then obviously $\log f_\epsilon=\min\{\log(1/\epsilon),\max\{\log(\epsilon),\log f\}\}$ and $\nabla \log f_\epsilon =(\nabla\log f)\chi_{[\epsilon<f<1/\epsilon]}$. Since after truncation we made sure that $\log f_\epsilon$ has bounded range we can use the composition with the exponential to conclude that $$ \nabla f_\epsilon=\nabla(\exp(\log f_\epsilon))=f_\epsilon\nabla \log f_\epsilon=(f\nabla\log f)\chi_{[\epsilon<f<1/\epsilon]}. $$ Because $f_\epsilon\to f$ in $L^1_{loc}$ we have $$ \int f_\epsilon\, div(\phi)\to \int f \,div(\phi) $$ for all test functions when $\epsilon\to 0$. On the other hand since $\nabla f_\epsilon=f_\epsilon\nabla\log f_\epsilon$ we can write $$ -\int f_\epsilon\, div(\phi)=\int f_\epsilon\nabla\log f_\epsilon\cdot \phi=\int (f\nabla\log f)\cdot(\phi\chi_{[\epsilon<f<1/\epsilon]}). $$ With your assumption that $\nabla\log f\in L^1(f\, dx)$ we can apply Lebesgue's dominated convergence to the last term (with the uniform bound almost everywhere $|(f\nabla\log f)\cdot(\phi\chi_{[\epsilon<f<1/\epsilon]})|\leq |f\nabla\log f|.\|\phi\|_{\infty}\in L^1$) and conclude that $$ \int (f\nabla\log f)\cdot(\phi\chi_{[\epsilon<f<1/\epsilon]})\to \int f\nabla\log f \cdot\phi. $$ Thus $$ \int f\,div(\phi)=-\int f\nabla\log f \cdot\phi $$ for all test-functions $\phi\in C^\infty_c$.