A priori estimates of Mizohata-Takeuchi type for the Navier-Lam\'e operator

TL;DR

This paper proves the Mizohata-Takeuchi conjecture for the Navier-Lamé operator in 2D and 3D, establishing weighted resolvent estimates near the spectrum with radial weights, extending known results for the Laplacian.

Contribution

It extends the Mizohata-Takeuchi estimates to the Navier-Lamé operator for radial weights, demonstrating limitations of previous methods based on maximal functions.

Findings

Proved the Mizohata-Takeuchi conjecture for Navier-Lamé in 2D and 3D.

Established that radial weights are not invariant under the Hardy-Littlewood maximal function.

Extended Laplacian estimates to the Navier-Lamé operator for specific weighted classes.

Abstract

The Mizohata-Takeuchi conjecture for the resolvent of the Navier-Lam\'e equation is a weighted estimate with weights in the so-called Mizohata-Takeuchi class for this operator when one approaches the spectrum (Limiting Absorption Principles). We prove this conjecture in dimensions 2 and 3 for weights with a radial majorant in the Mizohata-Takeuchi class. This result can be seen as an extension of the analogue for the Laplacian given in [8]. We also prove that radial weights in this class are not invariant for the Hardy-Littlewood maximal function, hence the methods in [6] used to extend estimates for the Laplacian to the Navier-Lam\'e case, do not work.