Translation-Invariant Estimates for Operators with Simple Characteristics

Abstract

We prove $L^{2}$ estimates and solvability for a variety of simply characteristic constant coefficient partial differential equations $P(D)u=f$. These estimates \[||u||_{L^2(D_{r})}\le C\sqrt{d_{r}d_{s}} ||f||_{_{L^2(D_{s})}}\] depend on geometric quantities - the diameters $d_{r}$ and $d_{s}$ of the regions $D_{r}$, where we estimate $u$, and $D_{s}$, the support of $f$ - rather than weights. As these geometric quantities transform simply under translations, rotations, and dilations, the corresponding estimates share the same properties. In particular, this implies that they transform appropriately under change of units, and therefore are physically meaningful. The explicit dependence on the diameters implies the correct global growth estimates. The weighted $L^{2}$ estimates first proved by Agmon in order to construct the generalized eigenfunctions for Laplacian plus potential in $\mathbb{R}^{n}$, and the more general and precise Besov type estimates of Agmon and H\"ormander, are all simple direct corollaries of the estimate above.