A generalized Poincaré inequality for a class of constant coefficient differential operators

Gustafson, Derek

doi:10.1090/S0002-9939-2011-10607-5

A generalized Poincaré inequality for a class of constant coefficient differential operators
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Proc. Amer. Math. Soc. 139 (2011), 2721-2728 Request permission

Abstract:

We study first order differential operators $\mathcal {P} = \mathcal {P}(D)$ with constant coefficients. The main question is under what conditions the following full gradient $L^p$ estimate holds: \[ \|D(f-f_0)\|_{L^p} \leq C \|\mathcal {P} f\|_{L^p},\ \textrm {for some } f_0 \in \ker \mathcal {P}.\] We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.

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