Structure constants of the Weyl calculus

TL;DR

This paper establishes explicit bounds on the operator norm of pseudo-differential operators in the Weyl calculus, showing dependence only on metric structure constants and symbol semi-norms, with implications for the Fefferman-Phong inequality.

Contribution

It provides new explicit bounds for operator norms in the Weyl calculus based on metric structure constants and symbol semi-norms.

Findings

Operator norm bounds depend only on structure constants and semi-norms.

Results apply to the Fefferman-Phong inequality.

Explicit bounds improve understanding of pseudo-differential operator behavior.

Abstract

We find some explicit bounds on the ${\mathcal L}(L^2)$-norm of pseudo-differential operators with symbols defined by a metric on the phase space. In particular, we prove that this norm depends only on the "structure constants" of the metric and a fixed semi-norm of the symbol. Analogous statements are made for the Fefferman-Phong inequality.