Locality and Domination of Semigroups

  • Khalid Akhlil


We characterize all semigroups \((T(t))_{t\ge 0}\) on \(L^2(\Omega )\) sandwiched between Dirichlet and Neumann ones, i.e.:
$$\begin{aligned} e^{t\Delta _D}\le T(t)\le e^{t\Delta _N},\quad \text {for all }t\ge 0 \end{aligned}$$
in the positive operators sense. The proof uses the well-known Beurling–Deny and Lejan formula to drop the locality assumption made usually on the form associated with \((T(t))_{t\ge 0}\). Moreover, we prove that if \(T(t)\le S(t)\) for all \(t\ge 0\), where \((T(t))_{t\ge 0}\) (resp. \((S(t))_{t\ge 0}\)) is a \(C_0\)-semigroup on some \(L^2\)-space associated with a regular Dirichlet form (aD(a)) (resp. with a Dirichlet form (bD(b))), then the locality of b implies the locality of a.


Robin boundary conditions locality domination of semigroups 

Mathematics Subject Classification

31C15 31C25 47D07 60H30 60J35 60J60 60J45 



The author would like to thank Wolfgang Arendt Omar El-Mennaoui and Jochen Glück for many stimulating and helpful discussions. This work was achieved during a research stay in Ulm, Germany. Many thanks to the referee who helped a lot to improve the first version of this paper.


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Authors and Affiliations

  1. 1.Applied Analysis InstituteUniversity of UlmUlmGermany
  2. 2.Polydisciplinary Faculty of OuarzazateIbn Zohr UniversityOuarzazateMorocco

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