A direct approach to the Weiss conjecture for bounded analytic semigroups

Abstract

Abstract. We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded H ∞ -calculus and is based on elementary analysis.

The Orlicz Weiss conjecture

“…In [4] the above lemma was done for Φ(t) = t 2 and our proof of "(ii) ⇒ (iii) ⇒ (i)" is based on the proof from Haak [4]. In [7] the equivalence of (i) and (ii) was shown for Φ(t) = t p and our proof relies on the ideas of [7].…”

“…He used generalized square function estimates for the operator A which are equivalent to (−A) 1 /p being infinite-time L p -admissible. Later, in [7] an, at the first glance, different proof for the statement of Haak was given. Since L p -admissibility, the p-Weiss conjecture and L p -admissibility of (−A) 1 /p are invariant under scaling of the semigroup and hence under shifting of the generator (for the latter see [15,Prop.…”

“…It is a nontrivial challenge to generalize these square function estimates for Orlicz spaces. Therefore, our approach is based on the slightly more elementary ideas from [7], even thought Haak's approach seems more natural.…”

“…Theorem 3.10 generalizes [4, Thm. 2.3] and [7,Lem 2.3]. In [4] the above lemma was done for Φ(t) = t 2 and our proof of "(ii) ⇒ (iii) ⇒ (i)" is based on the proof from Haak [4].…”

“…The approach in [7] uses the boundedness of the integral operator L, defined for τ > 0 by Lemma 3.12. If Φ ∈ P and L is given by (17) for some τ > 0 then L is a bounded operator on L Φ (0, τ ; Y ) with operator norm independent of τ > 0.…”