An Optimal Carleman-Type Inequality for the Dirac Operator

Abstract

We prove an inequality of the type

$$ \parallel {{e}^{{\tau \varphi }}}{\text{f}}\parallel {{{\text{ }}}_{{\text{L}}}}{{{\text{q}}}_{{{\text{(U)}}}}}{\text{ }} \leqq {\text{ c }}{{\left\| {{\text{ }}{{{\text{e}}}^{{\tau \varphi }}}\mathop{{\alpha D}}\limits^{{ \to \to }} {\text{f}}} \right\|}_{{{\text{ L}}}}}{{{\text{2}}}_{{{\text{(U)}}}}} $$

for τ → ∞, u ∈ C ^∞_O (U), with an optimal q. In particular, this gives the unique continuation property for Dirac operators $\mathop{{\alpha D}}\limits^{{ \to \to }} + V(x)$ when V ∈ L ^7/2_loc (IR ³). This result cannot be improved using isotropic L^P−L^q Carleman inequalities.