Hörmander condition with non-smooth $f$ Hörmander condition
Let $L$ denote the operator
\begin{equation}
L = \frac{1}{2}\sum\limits_{i=1}^k X_i ^2 + X_0,
\end{equation}
where $X_1,\dots,X_k$ are $C^\infty$ vector fields in $\mathbb{R}^n$. Assume that Lie Algebra generated by $X_1,\dots,X_k,X_0$ span $\mathbb{R}^n$. Then, if $u$ is a distribution such that
\begin{equation}
Lu = f
\end{equation}
and $f\in C^\infty$ in an open set G, then $u$ is $C^\infty$ in $G$.
My question is, what if $f$ is not $C^\infty$? In traditional results for parabolic equation, $f$ is only required to be Hölder continuous for some exponent $\alpha$, then $u$ can be shown to be $C^2$ with $\alpha$ Hölder derivatives. Does the same thing hold here?
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Hörmander's condition says that if $f$ is in a Sobolev space $H^s$, then under the condition on the vector fields then the solution is in a slightly better space. Take a look at Hörmander's 4 volumes here, here, here, and here.
Hörmander, Lars. The analysis of linear partial differential operators. I. 
Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256. Springer-Verlag, Berlin, 1983. ix+391 pp.
Hörmander, Lars. The analysis of linear partial differential operators. II. Differential operators with constant coefficients. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 257. Springer-Verlag, Berlin, 1983. viii+391 pp.
Hörmander, Lars. The analysis of linear partial differential operators. III. 
Pseudodifferential operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274. Springer-Verlag, Berlin, 1985. viii+525 pp.
Hörmander, Lars. The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275. Springer-Verlag, Berlin, 1985. vii+352 pp.
There are many papers about your problem.
If your vector fields are smooth, the regularities of solution to$$Lu = f$$have been studied by Rothschild and Stein (see Acta Mathematica 137 (1976) here) under various assumptions.
Rothschild, Linda Preiss; Stein, E. M. Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), no. 3-4, 247–320.
The paper is not easy to read and unfortunately the proof in the general case (when the vector fields $X_0$ is necessary for Hörmander's condition) is just sketched.
For "parabolic" operator generated using Hörmander's vector fields you can also check the following here.
Bramanti, Marco; Brandolini, Luca. Schauder estimates for parabolic nondivergence operators of Hörmander type. J. Differential Equations 234 (2007), no. 1, 177–245.
Also, I would suggest you the recent paper here and the references therein.
Bramanti, Marco; Zhu, Maochun. $L^p$ and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift. Anal. PDE 6 (2013), no. 8, 1793–1855. 
