Tensorization of quasiHilbertian Sobolev spaces
ErikssonBique, S., Rajala, T., & Soultanis, E. (2023). Tensorization of quasiHilbertian Sobolev spaces. Revista Matematica Iberoamericana, Online first. https://doi.org/10.4171/rmi/1433
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Revista Matematica IberoamericanaDate
2023Discipline
Analyysin ja dynamiikan tutkimuksen huippuyksikköMatematiikkaAnalysis and Dynamics Research (Centre of Excellence)MathematicsCopyright
© 2023 Real Sociedad Matemática Española
The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space X Y can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, W 1;2.X Y / D J 1;2.X; Y /, thus settling the tensorization problem for Sobolev spaces in the case p D 2, when X and Y are infinitesimally quasiHilbertian, i.e., the Sobolev space W 1;2 admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces X; Y of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally, for p 2 .1;1/ we obtain the normone inclusion kf kJ1;p.X;Y / kf kW 1;p.XY / and show that the norms agree on the algebraic tensor product W 1;p.X / ˝ W 1;p.Y / W 1;p.X Y /: When p D 2 and X and Y are infinitesimally quasiHilbertian, standard Dirichlet forms theory yields the density of W 1;2.X / ˝ W 1;2.Y / in J 1;2.X; Y /, thus implying the equality of the spaces. Our approach raises the question of the density of W 1;p.X / ˝ W 1;p.Y / in J 1;p.X; Y / in the general case.
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https://converis.jyu.fi/converis/portal/detail/Publication/183829120
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Academy of FinlandFunding program(s)
Academy Project, AoFAdditional information about funding
S. ErikssonBique was partially supported by the Finnish Academy grant no. 345005. T. Rajala was partially supported by the Finnish Academy grant no. 314789.License
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