New tensorial estimates in Besov spaces for time-dependent $(2 + 1)$-dimensional problems

TL;DR

This paper develops new tensorial estimates in geometric Besov spaces for evolving 2+1-dimensional surfaces, simplifying previous proofs and extending their applicability to more abstract settings, with applications in general relativity.

Contribution

It introduces simplified, more robust tensorial estimates in Besov spaces for time-dependent geometries, extending prior results to broader contexts.

Findings

Establishment of new tensorial estimates in Besov spaces

Simplification and robustness of proof techniques

Foundation for future applications in Einstein-vacuum spacetimes

Abstract

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodnianski, but in very specific settings. A primary objective of this paper is to significantly simplify and make more robust the proofs of the estimates. Another goal is to generalize these estimates to more abstract settings. In upcoming papers (joint with S. Alexakis), we will apply these estimates in order to study truncated null cones in an Einstein-vacuum spacetime extending to infinity. This analysis will then be used to study and to control the Bondi mass and the angular momentum under minimal conditions.