# Subelliptic Cordes estimates

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The classical Friedrichs identity states that for $u\in C_0^{\infty}(\mathbb{R}^n)$ we have

$$\int_{\mathbb{R}^n} |D^2 u|^2\, dx = \int_{\mathbb{R}^n} |\Delta u|^2\, dx.$$

From this inequality we immediately get $W^{2,2}$-estimates for

solutions of $\Delta u =f$ and also for solutions of measurable perturbations

of the form $\sum_{ij}a_{ij}(x) u_{ij}(x)=f(x)$, when the matrix

$A=(a_{ij})$ is closed to the identity in sense made precise

by Cordes.

In this talk we first explore extensions of the Friedrichs identity in

the form of sharp inequalities

$$\int_{X} |\mathfrak{X}^2 u|^2\, dx \le C_1 \int_{X} |\Delta_{\mathfrak{X}} u|^2\, dx +C_2\int_{X} |\mathfrak{X}u|^2 \, dx $$

where $X$ is a Riemannian manifold, the Heisenberg group, and certain types of CR manifolds.

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We then show how to use these estimates to study quasilinear subelliptic equations.\par

This is joint work with Andr\`as Domokos (PAMS 133, 2005) and Sagun

Chanillo (2007 preprint.)