Unique continuation for the heat operator with potentials in weak spaces

Abstract

We prove strong unique continuation property for the differential inequalityIn particular, we establish the strong unique continuation property for V ∈, which has been left open since the works of Escauriaza [6] and Escauriaza-Vega [8]. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.The sucp for the Laplacian −∆ is better understood. Since the pioneering work of Carleman [3], most of subsequent results were obtained by following his idea, the Carleman weighted inequality. In particular, Jerison and Kenig [13] proved the sucp for the Laplacian with V ∈ L d/2 loc , d ≥ 3. Their result was extended by Stein [28] to potentials V ∈ L d/2,∞ under the assumption that V L d/2,∞ is small enough. Later, Wolff [33] showed that the smallness assumption is indispensable if V ∈ L d/2,∞ . Here, • p,r denotes the norm of the Lorentz space L p,r (R d ) (for example, see [29]).

Endpoint eigenfunction bounds for the Hermite operator

“…See, Askey and Wainger [1], Karadzhov [9], and Thangavelu [23] (also, see [2,3] and [15] for recent development). The bound independent of λ has applications to the strong unique continuation problem for the parabolic operators [4,5,12,7].…”