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Tensor products and sums of $ p\mspace{1mu}$-harmonic functions, quasiminimizers and $ p\mspace{1mu}$-admissible weights


Authors: Anders Björn and Jana Björn
Journal: Proc. Amer. Math. Soc. 146 (2018), 5195-5203
MSC (2010): Primary 31C45; Secondary 35J60, 46E35
DOI: https://doi.org/10.1090/proc/14170
Published electronically: September 10, 2018
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Abstract: The tensor product of two $ p\mspace {1mu}$-harmonic functions is in general not $ p\mspace {1mu}$-harmonic, but we show that it is a quasiminimizer. More generally, we show that the tensor product of two quasiminimizers is a quasiminimizer. Similar results are also obtained for quasisuperminimizers and for tensor sums. This is done in weighted $ \mathbf {R}^n$ with $ p\mspace {1mu}$-admissible weights. It is also shown that the tensor product of two $ p\mspace {1mu}$-admissible measures is $ p\mspace {1mu}$-admissible. This last result is generalized to metric spaces.


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Additional Information

Anders Björn
Affiliation: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Email: anders.bjorn@liu.se

Jana Björn
Affiliation: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Email: jana.bjorn@liu.se

DOI: https://doi.org/10.1090/proc/14170
Keywords: Doubling measure, metric space, $p\mspace{1mu}$-admissible weight, $p\mspace{1mu}$-harmonic function, Poincar\'e inequality, quasiminimizer, quasisuperminimizer, tensor product, tensor sum
Received by editor(s): July 6, 2017
Received by editor(s) in revised form: March 20, 2018
Published electronically: September 10, 2018
Additional Notes: The authors were supported by the Swedish Research Council, grants 621-2007-6187, 621-2008-4922, 621-2014-3974, and 2016-03424.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2018 American Mathematical Society

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