Gerd Grubb - Microlocal and Global Analysis, In...
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Gerd Grubb - Microlocal and Global Analysis, Interactions with Geometry
Fractional-order operators and transmission spaces.
For a strongly elliptic pseudodierential operator P of order 2a (generally noninteger) | for example
the fractional Laplacian ()a, 0 < a < 1 | there is a useful theory for boundary value problems on smooth subsets
of Rn, when P satises Hormander's a-transmission condition at the boundary. For the homogeneous Dirichlet
problem (u supported in
), the solution space for data in Hs(
), s 0, is then the a-transmission space
) = (a)
), where (t)
+ denotes an order-reducing pseudodierential operator of plus-type. A
similar result holds in Holder-Zygmund spaces. We shall give a more down-to-earth description of the transmission
spaces, involving a power da of the distance d(x) = dist(x; @
) and a Poisson solution operator for the Laplacian.
There are nice