Carleman Inequalities and the Heat Operator

1. Introduction. The unique continuation property is best understood for second-order elliptic operators. The classic paper by Carleman [8] established the strong unique continuation theorem for second-order elliptic operators that need not have analytic coefficients. The powerful technique he used, the so-called " Carleman weighted inequality, " has played a central role in later developments. In the 1950s, Aronszajn [3] and Cordes [11] generalized Carleman's result to higher dimensions. In recent years, this subject attracted attention from a great number of people. Many efforts have been made to relax the smoothness hypothesis on the coefficients (see [7], [15], [31], [2], and [16]). Using this method, Jerison and Kenig [19] obtained the strong unique continuation property for operators of the form + V with V ∈ L n/2 loc , n ≥ 3. Further improvements have been made in considering other classes of coefficients (Fefferman-Phong class and Kato class) (see [30], [9], [12], and [24]), in extending the result to operators with first derivative terms and variable coefficients the references therein). For second-order linear parabolic operators with time-independent coefficients, the strong unique continuation property was reduced in [25] to the previously established elliptic counterparts. In particular, it is shown in [25] that if u is a solution of u + ∂ t u + V (x)u = 0 in S T = × (0, T), V ∈ L (n+1)/2 ((), (x 0 , t 0) ∈ S T , and B r (x 0) u 2 (x, t 0) dx ≤ C N r N for any integer N. Then u(x, t 0) ≡ 0 for x ∈ , and in addition, assuming that u = 0 on ∂∂ × (0, T), then u ≡ 0 in S T. The reduction from time-independent parabolic equations to elliptic equations, a basic technique used in [25], relies on a representation formula for solutions of para-bolic equations in terms of eigenfunctions of the corresponding elliptic operator, and therefore cannot be applied to more general equations with time-dependent coefficients (for the weak unique continuation, see also [17], [41]). Time-dependent parabolic equations with variable coefficients are treated in [29], [34], where a weak unique continuation theorem is proven using a Carleman inequality. In [29] this was established for variable C 2 second-order coefficients and bounded first-and zero-order terms, while in [34] unbounded time-dependent potentials were treated. In particular, it is shown that if u verifies ||u …