The Weiss conjecture and weak norms

TL;DR

This paper demonstrates the equivalence between the Weiss condition and weak Lebesgue admissibility for analytic semigroups, establishing the optimality of the weak Lebesgue norm as the endpoint in Lorentz spaces for the Weiss conjecture.

Contribution

It proves the equivalence of Weiss condition and weak Lebesgue admissibility for analytic semigroups and shows the weak Lebesgue norm's optimality in Lorentz spaces.

Findings

Weiss condition and weak Lebesgue admissibility are equivalent for analytic semigroups.

Weak Lebesgue norm is the endpoint in Lorentz spaces for the Weiss conjecture.

The result clarifies the optimality of the weak Lebesgue norm in this context.

Abstract

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 $L^{p,q}$.