Due to a theorem of Malliavin (an improvement of an erlier theorem of Dixmier and Malliavin, see here) we know that every compactly supported smooth function $f$ on $\mathbb R^n$ can be written as a sum of two convolutions $$f=u_1\circ v_1+u_2\circ v_2$$ where the $u_i,v_i$ are smooth.

Is it known whether we can choose the $u_i,v_i$ to vary continuously with $f$?

Due to a theorem of Malliavin (an improvement of an erlier theorem of Dixmier and Malliavin, see here) we know that every compactly supported smooth function $f$ on $\mathbb R^n$ can be written as a sum of two convolutions $$f=u_1\circ v_1+u_2\circ v_2$$ where the $u_i,v_i$ are smooth.

Question: Is it known whether we can choose the $u_i,v_i$ to vary continuously with $f$?

The original theorem By Dixmier and Malliavin proves that $f$ is the sum of finitely many convolutions (I believe they prove that $2^n$ is enough). In this case we can choose the functions to depend continuously on $f$, as is proved in this undergraduate thesis.

Malliavin's improvement of the theorem appears in "$C^{\infty}$ parametrix on Lie groups and two steps factorization on convolution algebras", Harmonic analysis (Proc. Conf., Univ. Crete, Iraklion, 1978), pp. 142–156, Lecture Notes in Math., 781, Springer, Berlin, 1980.