The Weiss conjecture and weak norms

Bernhard Hermann Haak (1)

1 IMB - Institut de Mathématiques de Bordeaux

In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators \[ Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0 \] on the complex right half plane and weak Lebesgue $L^{2,\infty}$--admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$.

Mots clés: Lorentz spaces; Weiss conjecture; Observation of linear systems

Domaines: Analyse fonctionnelle [math.FA]; Optimisation et contrôle [math.OC]

Bernhard Hermann Haak. The Weiss conjecture and weak norms. Journal of Evolution Equations, 2012, 12 (4), pp.855-861. ⟨hal-00700159v2⟩