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For an elliptic operator $$ Lu = (a^{ij} D_iD_j + b^i D_i + c)u = f,$$ with suitable assumptions on the coefficients, one usually has Schauder estimates of the form $$ \|u\|_{C^{2, \alpha}} \leq C(\|f …

You are done once you know that you have a functional calulus for the Laplace-Beltrami operator on $M$. For this, show that it is self-adjoint and has nonpositive spectrum (there are various ways to d …

Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$? For example, is the dimension of $V$ alw …

The answer to this question as asked is no. However, you generally obtain something similar. Consider $D = Q + Q^*$. By standard arguments, $$\ker(D) = \ker(Q)\cap \ker(Q^*).$$ (Of course $D\Phi = 0$ …

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function …

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution $\mathc …