×

New proofs of the trace theorem of Sobolev spaces. (English) Zbl 1170.46034

Summary: We present three new proofs of the trace theorem for \(L_{p}\) Sobolev spaces, which do not rely on the theory of interpolation spaces. The first method originates in Morrey’s proof of the Sobolev embedding theorem concerning the Hölder-Zygmund space. The second method is based on Muramatu’s integral formula and the third method is based on an integral operator with Gauss kernel. These methods give unified viewpoints for the proofs of the trace theorem and the Sobolev embedding theorem.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] R. A. Adams and J. F. Fournier, Sobolev spaces , 2nd edition, Pure and Applied Mathematics, 140, Academic Press, Amsterdam, 2003. · Zbl 1098.46001
[2] J. Bergh and J. Löfström, Interpolation spaces. An introduction , Grundlehren der Mathematischen Wissenschaften, 223, Springer, Berlin-New York, 1976.
[3] V. I. Burenkov, Sobolev spaces on domains , Teubner, Stuttgart, 1998.
[4] E. DiBenedetto, Real analysis , Birkhäuser Boston, Boston, MA, 2002. · Zbl 1012.26001
[5] C. B. Morrey, Jr., Multiple integrals in the calculus of variations , Springer-Verlag New York, Inc., New York, 1966. · Zbl 0142.38701
[6] T. Muramatu, On Besov spaces and Sobolev spaces of generalized functions definded on a general region, Publ. Res. Inst. Math. Sci. 9 (1973/74), 325-396. · Zbl 0287.46046
[7] E. M. Stein, The characterization of functions arising as potentials II, Bull. Amer. Math. Soc. 68 (1962), 577-582. · Zbl 0127.32002
[8] H. Triebel, Interpolation theory, function spaces, differential operators , North-Holland Mathematical Library, 18, North-Holland, Amsterdam-New York, 1978. · Zbl 0387.46032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.