Suppose that $\Delta f\in L^1(\mathbb{R}^n)$, where $\Delta$ denotes the Laplacian. Then I’m asking whether or not we have the following estimates $$ \|\nabla|f|^{\frac12}\|_{L^2}\leq C\|\Delta f\|_{L^1}^{\frac12}. $$ i.e., does the mapping $f\to |f|^{\frac12}$ send functions from $\dot{W^{2,1}}$ to $\dot{W^{1,2}}$ ?

It seems that a paper of W. Sickel (Boundedness properties of the mapping $f\to |f|^{\mu}$, $0 <\mu< 1$ in the framework of Besov spaces) has shown that the inhomogeneous version of the inequality above is indeed true. But I'm not able to get that paper for now, and more importantly, I'm not sure whether the method there will also work for the case I'm concerning.

Any comment is welcome.

Suppose that $\Delta f\in L^1(\mathbb{R}^n)$, where $\Delta$ denotes the Laplacian. Then I’m asking whether or not we have the following estimates $$ \|\nabla|f|^{\frac12}\|_{L^2}\leq C\|\Delta f\|_{L^1}^{\frac12}. $$ i.e., does the mapping $f\to |f|^{\frac12}$ send functions from $\dot{W^{2,1}}$ to $\dot{W^{1,2}}$ ?

It seems that a paper of W. Sickel (Boundedness properties of the mapping $f\to |f|^{\mu}$, $0 <\mu< 1$ in the framework of Besov spaces) has shown that the inhomogeneous version of the inequality above is indeed true. However I'm not able to get that paper for now, and more importantly, I'm not sure whether the method there will also work for the case I'm concerning.

Any comment is welcome.