×

Sobolev type embeddings in the limiting case. (English) Zbl 0930.46027

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) and let \(k\in\mathbb{N}\). The limiting embedding \[ W^k_p(\Omega)\subset L_\Phi(\Omega,\mu),\quad p={n\over k}, \] in some Orlicz spaces \(L_\Phi\), where \(\mu\) is a suitable measure, attracted a lot of attention since the late sixties. The paper contributes to this subject, generalizing the source space by \(W^k_A\), where \(A\) is a rearrangement-invariant space near the above space \(L_p\), and the optimal target spaces are described in terms of rearrangement, extending and sharpening what is known so far.
Reviewer: H.Triebel (Jena)

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.)
46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Adams, D.R. and Hedberg, L.I. (1966).Function Spaces and Potential Theory, Springer-Verlag, Berlin. · Zbl 0834.46021
[2] Beckenbach, E.F. and Bellman, R. (1961).Inequalities, Springer-Verlag, Berlin.
[3] Bennett, C. and Rudnick, K. (1980) On Lorentz-Zygmund spaces,Dissertationes Math.,175, 5–67. · Zbl 0456.46028
[4] Bennett, C. and Sharpley, R. (1988).Interpolation of Operators. Academic Press, New York. · Zbl 0647.46057
[5] Bergh J. and Löfström, J. (1976).Interpolation Spaces. An Introduction, Springer-Verlag, Berlin. · Zbl 0344.46071
[6] Boyd, D.W. (1969). Indices of function spaces and their relationship to interpolation,Canad. J. Math.,21, 1245–1254. · Zbl 0184.34802 · doi:10.4153/CJM-1969-137-x
[7] Brezis, H. and Wainger, S. (1980). A note on limiting cases of Sobolev embeddings.Comm. Partial Diff. Equations,5, 773–789. · Zbl 0437.35071 · doi:10.1080/03605308008820154
[8] Brudnyi, Ju.A. (1979). Rational approximation and imbedding theorems,Dokl. Akad. Nauk SSSR,247, 269–272; English translation inSov. Math. Dokl.,20, 681–684.
[9] Cwikel, M. and Pustylnik, E. (1998). Weak type interpolation near ”endpoint”, spaces. Preprint. · Zbl 0978.46008
[10] Edmunds, D.E., Gurka, P., and Opic, B. (1995). Double exponential, integrability, Bessel potentials and embedding theorems,Studia Math.,115, 151–181. · Zbl 0829.47024
[11] Hansson, K. (1979). Imbedding theorems of Sobolev type in potential theory,Math. Scand.,45, 77–102. · Zbl 0437.31009
[12] Krasnoselskii, M.A., Zabreiko, P.P., Pustylnik, E.I., and Sobolevskii, P.E. (1966).Integral Operators in Spaces of Summable Functions, Izd. Nauk, Moscow, English translation, Noordhoff, Leyden (1976).
[13] Krein, S.G., Petunin, Ju.I., and Semenov, E.M. (1978).Interpolation of Linear Operators, Izd. Nauka, Moscow, English translation in Translations of Mathem. Monographs, Vol. 54,American Math. Soc., Providence RI, (1982).
[14] Lorentz, G.G. (1950). Some new functional spaces,Ann. Math.,51, 37–55. · Zbl 0035.35602 · doi:10.2307/1969496
[15] Maz’ja, V.G. (1985).Sobolev Spaces, Springer-Verlag, Berlin.
[16] Peetre, J. (1966). Espaces d’interpolation et théorème de Soboleff,Ann. Inst. Fourier,16, 279–317. · Zbl 0151.17903
[17] Sobolev, S.L. (1938). On a theorem of functional analysis,Mat. Sbornik,4(46), 471–497; English translation inAmer. Math. Soc. Transl.,34, 39–68. (1963).
[18] Sobolev, S.L. (1963). Applications of functional analysis in mathematical physics, Vol. 7, Transl. of Math. Monographs,American Math. Soc., Providence, RI. · Zbl 0123.09003
[19] Stein, E.M. (1970).Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ. · Zbl 0207.13501
[20] Strichartz, R.S. (1972). A note on Trudinger’s extension of Sobolev’s inequality,Indiana U. Math.,21, 841–842. · Zbl 0241.46028 · doi:10.1512/iumj.1972.21.21066
[21] Triebel, H. (1983).Theory of Function Spaces, Birkhäuser, Basel. · Zbl 0546.46028
[22] Trudinger, N. (1967). On embeddings into Orlicz spaces and some applications.J. Math. Mech.,17, 473–483. · Zbl 0163.36402
[23] Yano, S. (1951). An extrapolation theorem.J. Math. Soc. Japan,3, 296–305. · Zbl 0045.17901 · doi:10.2969/jmsj/00320296
[24] Zygmund, A. (1968).Trigonometric Series, Vol. II, Cambridge University Press. · Zbl 0157.38204
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.