An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces. (English) Zbl 0913.46028
An extension of the Marcinkiewicz weak interpolation theorem for quasilinear operators to the case when the spaces generating the interpolation are Orlicz spaces is proved. The result requires some auxiliary functions \(E_{\Phi,\beta}(s)\), \(F_{\Phi,\beta}(s)\), \(G_{\Phi,\beta}(s)\) and \(H_{\Phi,\beta}(s)\) depending on an Orlicz function \(\Phi\) and is obtained by means of real methods.
Reviewer: J.Musielak (Poznań)
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46M35 | Abstract interpolation of topological vector spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42A65 | Completeness of sets of functions in one variable harmonic analysis |
| 46N20 | Applications of functional analysis to differential and integral equations |
Keywords:
Marcinkiewicz weak interpolation theorem; quasilinear operators; Orlicz spaces; Orlicz functionReferences:
| [1] | Bennett, C.; Rudnick, K., On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.), CLXXV (1980) · Zbl 0456.46028 |
| [2] | Bennett, C.; Sharpley, R., Interpolation of Operators (1988), Academic Press: Academic Press San Diego · Zbl 0647.46057 |
| [3] | Calderón, A. P., Spaces between \(L^1L^∞\), Studia Math., 26, 273-299 (1966) · Zbl 0149.09203 |
| [4] | Cianchi, A., Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23, 575-608 (1996) · Zbl 0877.46023 |
| [5] | A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc.; A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. · Zbl 0940.46015 |
| [6] | A. Cianchi, Hardy inequalities in Orlicz spaces, Trans. Amer. Math. Soc.; A. Cianchi, Hardy inequalities in Orlicz spaces, Trans. Amer. Math. Soc. · Zbl 0920.46021 |
| [7] | Edmunds, D. E.; Gurka, P.; Opic, B., Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J., 44, 19-43 (1995) · Zbl 0826.47021 |
| [8] | Evans, W. D.; Opic, B.; Pick, L., Interpolation of operators on scales of generalized Lorentz-Zygmund spaces, Math. Nachr., 182, 127-181 (1996) · Zbl 0865.46016 |
| [9] | Fusco, N.; Lions, P. L.; Sbordone, C., Some remarks on Sobolev imbeddings in borderline cases, Proc. Amer. Math. Soc., 124, 561-565 (1996) · Zbl 0841.46023 |
| [10] | Gustavsson, J.; Peetre, J., Interpolation of Orlicz spaces, Studia Math., 60, 33-59 (1977) · Zbl 0353.46019 |
| [11] | Hunt, R. A., An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces, Bull. Amer. Math. Soc., 70, 803-807 (1964) · Zbl 0145.38203 |
| [12] | Kraynek, W. T., Interpolation of sublinear operators in generalized Orlicz and Hardy-Orlicz spaces, Studia Math., 43, 93-123 (1972) · Zbl 0235.46055 |
| [13] | Marcinkiewicz, J., Sur l’interpolation d’operateurs, C. R. Acad. Sci. Paris, 1272-1273 (1939) · JFM 65.0506.03 |
| [14] | Maz’ja, V. M., Sobolev Spaces (1985), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0692.46023 |
| [15] | O’Neil, R., Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc., 115, 300-328 (1965) · Zbl 0132.09201 |
| [16] | Peetre, J., A new approach in interpolation spaces, Studia Math., 34, 23-42 (1970) · Zbl 0188.43602 |
| [17] | Pustyl’nik, E. I., On optimal interpolation and some interpolation properties of Orlicz spaces, Soviet Math. Dokl., 27 (1983) · Zbl 0539.46050 |
| [18] | Rao, M. M., Interpolation, ergodicity and martingales, J. Math. Mech., 16, 543-567 (1966) · Zbl 0166.11704 |
| [19] | Rao, M. M.; Ren, Z. D., Theory of Orlicz Spaces (1991), Dekker: Dekker New York · Zbl 0724.46032 |
| [20] | Riordan, W. J., On the Interpolation of Operators (1956), University of Chicago |
| [21] | Sawyer, E. T., Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96, 145-158 (1990) · Zbl 0705.42014 |
| [22] | Strichartz, R. S., A note on Trudinger’s extension of Sobolev’s inequalities, Indiana Univ. Math. J., 21, 841-842 (1972) · Zbl 0241.46028 |
| [23] | Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3, 697-718 (1976) · Zbl 0341.35031 |
| [24] | Torchinsky, A., Interpolation of operators and Orlicz classes, Studia Math., 59, 177-207 (1976) · Zbl 0348.46027 |
| [25] | Torchinsky, A., Real Variable Methods in Harmonic Analysis (1986), Academic Press: Academic Press San Diego · Zbl 0621.42001 |
| [26] | Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17, 473-483 (1967) · Zbl 0163.36402 |
| [27] | Zygmund, A., On a theorem of Marcinkiewicz concerning interpolation of operations, J. Math. Pures Appl., 35, 223-248 (1956) · Zbl 0070.33701 |
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