WG4 Research Seminar - Talk by Jan Meichsner
On February 9th, 2022, Jan Meichsner (FernUniversität in Hagen) will give a talk at 3:30 pm CET about
"Fractional powers of linear operators and the Caffarelli-Silvestre extension".
The talk will be held online in Zoom as part of the research seminar Analysis of the FernUniversität in Hagen. The zoom URL will be sent to all members of the WG4 and to all further colleagues who send a mail to Delio Mugnolo.
Abstract
The presentation will have two parts: first we start with a rather quick introduction into the well-studied field of fractional operators on Banach spaces, point out the important ideas, and mention some results. We then continue with a possibility to compute them going back to [1] where the authors attracted a lot of attention when applying their techniques to a well-known special instance of fractional operator, the fractional Laplacian, whose action on a function f can be described using a solution of a certain ODE. One may generalise to arbitrary sectorial operators A instead of the Laplacian and consider the ODE in some general Banach space X. Many other authors [2, 3, 4] contributed to this more general framework. The talk answers the questions whether the considered ODE always has a unique solution and to what extend this solution can be used to describe fractional powers of A. Details are available in [5, 6] as well as in [7].
Literatur
[1] L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32(8): 1245–1260, 2007.
[2] P. R. Stinga and J. L. Torrea. Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differential Equations, 35(11): 2092–2122, 2010.
[3] J. E. Galé, P. J. Miana and P. R. Stinga. Extension problem and fractional operators: semigroups and wave equations. J. Evol. Equ., 13(2): 343–368., 2013.
[4] W. Arendt, A. F. M. ter Elst and M. Warma. Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Comm. Partial Differential Equations, 43(1): 1–24, 2016.
[5] J. Meichsner, C. Seifert. On the harmonic extension approach to fractional powers in Banach spaces. Fract. Calc. Appl. Anal., 23(4): 1054–1089, 2020.
[6] J. Meichsner, C. Seifert. On some Consequences of the Solvability of the Caffarelli–Silvestre Extension Problem. In Bastos M.A, Castro L., Karlovich A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications., vol 282. Birkhäuser, Cham., 2020.
[7] J. Meichsner. Fractional Powers of Linear Operators in Locally Convex Vector Spaces. Ph.D. thesis, Technische Universität Hamburg (2021) https://tore.tuhh.de/bitstream/11420/9944/3/Meichsner_Jan_Fractional-Powers-of-Linear-Operators-in-Locally-Convex-Vector-Spaces.pdf
