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

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.