×

Landau type theorem for Orlicz spaces. (English) Zbl 0738.46013

This paper contains a short and elementary proof of the following fact: If \(L^ m\) is an Orlicz space generated by a convex (not necessarily finite-valued) function \(M\) and \(g\) is a measurable function such that \(fg\in L^ 1\) for all \(f\in L^ M\), then \(g\in L^{M*}\), where \(M^*\) is the conjugate (=complementary) function of \(M\).
This proposition implies analogous theorems for some classes of nonlocally convex Orlicz spaces (over atomless or counting measures). Moreover, this paper contains an example of an Orlicz space \(L^ M\), whose Köthe dual is not isomorphic to any space of the form \(L^{M*}(\nu)\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Abramoviĉ, Ju. A., Wojtaszczyk, P.: The uniqueness of the order in spacesL p [0, 1] andl p (in Russina). Mat. Zametki18, 313–325 (1975)
[2] Barbu, V., Precupanu, Th.: Convexity and Optimization of Banach Spaces. Bucharest: Sijthoff and Noordhoff 1978 · Zbl 0379.49010
[3] Birnbaum, Z.W., Orlicz, W.: Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen. Stud. Math.3, 1–67 (1931); reprinted in: Orlicz, W.: Collected Papers, pp. 133–199. Warsaw: Polish Scientific Publishers 1988 · Zbl 0003.25202
[4] Drewnowski, L.: Compact operators on Musielak-Orlicz spaces. Ann. Soc. Math. Pol., Ser. I, Commentat. Math.27, 225–232 (1988) · Zbl 0676.46024
[5] Drewnowski, L., Nawrocki, M.: On the Mackey topology of Orlicz sequence spaces. Arch. Math.39, 59–68 (1982) · Zbl 0502.46004
[6] Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems (in Russian). Moscow: Nauka 1974
[7] Kantoroviĉ, L.V., Akilov, G.P.: Functional Analysis (in Russian). Moscow: Nauka 1984
[8] Krasnoesel’skii, M.A., Rutickii, Ya.B.: Convex Functions and Orlicz Spaces. Groningen: Noordhoff 1961
[9] Landau, E.: Über einen Konvergenzsatz. Nachr. Königl. Ges. Wiss. Göttingen Nr.8, 25–27 (1907) · JFM 38.0296.01
[10] Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Indagationes Math.51, 323–338 (1989) · Zbl 0704.46018
[11] Musielak, J.: Orlicz Spaces and Modular Spaces (Lect Notes Math., vol. 1034). Berlin Heidelberg New York: Springer 1983 · Zbl 0557.46020
[12] Nowak, M.: The Köthe dual of Orlicz sequence spaces without local convexity. Math. Japonica34, 619–627 (1989) · Zbl 0684.46012
[13] Wnuk, W.: Orlicz spaces with are Riesz isomorphic tol Rocky Mt. J. Math.18, pp. 185–193 (1988) · Zbl 0665.46020
[14] Zaanen, A.C.: Riesz Spaces II. Amsterdam New York Oxford: North-Holland 1983 · Zbl 0519.46001
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.