Duality of ODE-determined norms
DOI:
https://doi.org/10.7146/math.scand.a-109390Abstract
Recently the author initiated a novel approach to varying exponent Lebesgue space $L^{p(\cdot)}$ norms. In this approach the norm is defined by means of weak solutions to suitable first order ordinary differential equations (ODE). The resulting norm is equivalent with constant $2$ to a corresponding Nakano norm but the norms do not coincide in general and thus their isometric properties are different. In this paper the duality of these ODE-determined $L^{p(\cdot)}$ spaces is investigated. It turns out that the duality of the classical $L^p$ spaces generalizes nicely to this class of spaces. Here the duality pairing and Hölder's inequality work in the isometric sense which is a notable feature of these spaces. The uniform convexity and smoothness of these spaces are characterized under the anticipated conditions. A kind of universal space construction is also given for these spaces.
References
Coddington, E. A. and Levinson, N., Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., and Zizler, V., Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4757-3480-5
Jamison, J. E., Kamińska, A., and Lin, P.-K., Isometries of Musielak-Orlicz spaces. II, Studia Math. 104 (1993), no. 1, 75–89. https://doi.org/10.4064/sm-104-1-75-89
Kováčik, O. and Rákosník, J., On spaces $L^p(x)$ and $W^k,p(x)$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618.
Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973.
Luxemburg, W. A. J., Banach function spaces, Thesis, Technische Hogeschool te Delft, 1955.
Maligranda, L., Hidegoro Nakano (1909-1974)—on the centenary of his birth, in “Banach and function spaces III (ISBFS 2009)'', Yokohama Publ., Yokohama, 2011, pp. 99‒171.
Marcinkiewicz, J., Sur l'interpolation d'opérations, C. R. Acad. Sci., Paris 208 (1939), 1272–1273.
Musielak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0072210
Nakai, E. and Sawano, Y., Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748. https://doi.org/10.1016/j.jfa.2012.01.004
Orlicz, W., Über eine gewisse Klasse von Räumen vom Typus $B$, Bull. Int. Acad. Polon. Sci. A 1932 (1932), no. 8-9, 207–220.
Rao, M. M. and Ren, Z. D., Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991.
Rao, M. M. and Ren, Z. D., Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker, Inc., New York, 2002. https://doi.org/10.1201/9780203910863
Sobczyk, A., Projections in Minkowski and Banach spaces, Duke Math. J. 8 (1941), 78–106.
Talponen, J., A natural class of sequential Banach spaces, Bull. Pol. Acad. Sci. Math. 59 (2011), no. 2, 185–196. https://doi.org/10.4064/ba59-2-8
Talponen, J., Note on order-isomorphic isometric embeddings of some recent function spaces, J. Funct. Spaces (2015), Art. 186105, 6 pp. https://doi.org/10.1155/2015/186105
Talponen, J., ODE for $L^p$ norms, Studia Math. 236 (2017), no. 1, 63–83. https://doi.org/10.4064/sm8561-8-2016
Talponen, J., Decompositions of Nakano norms by ODE techniques, Ann. Acad. Sci. Fenn. Math. 43 (2018), 737–754.
