Translation-Invariant Estimates for Operators with Simple Characteristics

TL;DR

This paper establishes translation-invariant $L^{2}$ estimates for simply characteristic constant coefficient PDEs, linking solutions' norms to geometric diameters of regions, ensuring physically meaningful and scalable bounds.

Contribution

It introduces geometric diameter-based $L^{2}$ estimates for PDEs with simple characteristics, unifying and extending previous weighted and Besov estimates.

Findings

Estimates depend on geometric diameters, not weights.

Estimates are invariant under translations, rotations, and dilations.

Includes corollaries for classical weighted and Besov estimates.

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.