Harmonic and Trace Inequalities in Lipschitz Domains
We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore–Penrose inverse, and of a special inner product. We show that our trace inequalities are particularly useful to prove harmonic inequalities, which serve as powerful tools to characterize the harmonic functions on Sobolev spaces of non-integer order.
KeywordsMoore–Penrose equality Trace inequalities Harmonic inequalities Lipschitz domains Trace spaces Harmonic functions Hilbert spaces
2010 Mathematics Subject Classification47A30 47J20
This research is part of the first author’s Ph.D. project, which is carried out at Moulay Ismail University, Meknes. It was essentially finished during a visit of Touhami to the Department of Mathematics of University of Aveiro, Portugal, November 2018. The hospitality of the host institution and the financial support of Moulay Ismail University, Morocco, and CIDMA, Portugal, are here gratefully acknowledged. Torres was partially supported by the Portuguese Foundation for Science and Technology (FCT) through CIDMA, project UID/MAT/04106/2019.
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