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Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space: $$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...

I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. In this article, the authors proved the following theorem (Theorem 2.3) Let ...