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In this Note we give some compact embedding theorems for Sobolev spaces, related to $m$-tuples of vectors fields of $C^1$ class on ${\mathbb{R}}^N$.

Abbiamo considerato il problema della regolarità interna delle soluzioni deboli della seguente equazione differenziale $$\stackrel{m_0}{\sum _{i,j=1}}{\partial }_{x_i}\left(a_{i,j}\left(x,t\right){\partial }_{x_j}u\right)+\stackrel{N}{\sum _{i,j=1}}{b}_{i,j}{x}_i{\partial }_{x_j}u-{\partial }_{t}u=\stackrel{m_0}{\sum _{j=1}}{\partial }_{x_j}{F}_j\left(x,t\right),$$ dove $\left(x,t\right)\in {\mathbb{R}}^{N+1}$, $0<m_0\le N$ ed $F_j\in L_{\text{loc}}^p\left({\mathbb{R}}^{N+1}\right)$ per $j=1,\mathrm{\dots },m_0$. I nostri principali risultati sono una stima a priori interna del tipo $$\stackrel{m_0}{\sum _{j=1}}{∥{\partial }_{x_j}u∥}_p\le c\left(\stackrel{m_0}{\sum _{j=1}}{∥F_j∥}_p+{∥u∥}_p\right),$$ e la regolarità hölderiana di $u$. La stima a priori delle derivate viene ottenuta utilizzando una tecnica analoga a quella introdotta da Chiarenza, Frasca e Longo in [3], per gli operatori ellittici in forma di non divergenza, supponendo che i coefficienti $a_{i,j}$ verifichino una condizione...

In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a...