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Algebra properties for Sobolev spaces – applications to semilinear PDEs on manifolds. (English) Zbl 1286.46033
Some algebra properties are given for generalized Sobolev spaces \(W^{s,p}\cap L^\infty\) on a Riemannian manifold (or, more generally, a homogeneous type space), where \(W^{s,p}\) is a Bessel potential of the kind \((1+ L)^{-s/m}(L^p)\) and \(L\) is an operator generating a heat semigroup with off-diagonal decay. These results are extended to Riemannian manifolds with weaker geometrical conditions using the \(L^p\)-boundedness of the Riesz transformations. Two different approaches, namely Bony’s paraproducts and functionals, are used.
The authors claim that their results are new on Riemannian manifolds with an unbounded geometry and even for Sobolev spaces in the case of Euclidean spaces. Nonlinear transformations between Sobolev spaces are also studied and then applied to obtain well-posedness results for semilinear PDEs (Schrödinger and heat equations) with regular data. This extends previous work by Strichartz, Kato-Ponce and many other authors.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
47D03 Groups and semigroups of linear operators
47A60 Functional calculus for linear operators
58J05 Elliptic equations on manifolds, general theory
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