Weighted Estimates for the Helmholtz Equation and Some Applications

Decay estimates for the wave and Dirac equations with a magnetic potential

“…For A = −∆ this is a classical result (see e.g. Agmon [1]); here we shall use a very precise version of the principle, due to Barcelo, Ruiz and Vega [4]. On the other hand, for the Dirac operator only a few results are available, which concern the case with a nonzero mass term (see [28], [42]).…”

“…Remark 1.2. As an essential step in the proof of Theorem 1.1, we need to establish the limiting absorption principle (LAP) for the operator H. This is obtained in Section 3 through several steps: starting from the "weak" LAP of [4] for the free resolvent, we first prove a strong version of the LAP for the free operator in the weighted spaces…”

Some Recent Progress on the Restriction Conjecture

“…These three surfaces are model examples of hypersurfaces with curvature 4 , though of course the cone differs from the sphere and paraboloid in that it has one vanishing principal curvature. These three hypersurfaces also enjoy a large group of symmetries (the orthogonal group, the parabolic scaling and Gallilean groups, and the Poincare group, respectively).…”

“…For the PDE applications it is sometimes convenient to think of R n as a spacetime R n−1 × R := {(x, t) : x ∈ R n−1 , t ∈ R} (with the frequency space thus becoming spacetime frequency space {(ξ, τ ) : ξ ∈ R n−1 , τ ∈ R}), but we will avoid doing so here. 4 One could also consider cylinders such as S k−1 × R n−k ⊂ R n , but it turns out that the restriction theory for these surfaces is identical to that of the sphere S k−1 inside R k . 5 Indeed, it suffices for the weak-type estimate from L p (R n ) to L p,∞ (S; dσ) to fail.…”

“…We will not pursue these connections further here, but see for instance [49] and the (numerous) papers descended from that paper. (Some other connections between restriction estimates and PDEtype estimates are summarized in [62] and the references therein; for the Helmholtz equation, see for instance [4]. )…”