Abstract
Let\(\mathcal{H}^{1,1} (T^1 )\) denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space\(\mathcal{H}^1 (T^1 )\). It is shown that when the weak derivative of a locally Lipschitz continuous functionf has bounded variation on compact sets the Nemytskii operatorF, defined byF(u)=f·u, maps\(\mathcal{H}^{1,1} (T^1 )\) continuously into itself. A further condition sufficient for the continuous Fréchet differentiability ofF is then added.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Appell, J. andZabrejko, P. P.,Nonlinear Superposition Operators, Cambridge Tracts in Math.95, Cambridge Univ. Press, Cambridge, 1990.
Janson, S., On functions with derivatives inH 1, inHarmonic Analysis and Partial Differential Equations (El Escorial, 1987) (García-Cuerva, J., ed.), Lecture Notes in Math.1384, pp. 193–201, Springer-Verlag, Berlin-Heidelberg, 1989.
Marcus, M. andMizel, V. J., Every superposition operator mapping one Sobolev space into another is continuous,J. Funct. Anal. 33 (1979), 217–229.
Runst, T. andSickel, W.,Sobolev Spaces of Fractional Order, Nemytskii Operators and Nonlinear Partial Differential Equations, de Gruyter, Berlin, 1996.
Toland, J. F., Regularity of Stokes waves in Hardy spaces and as distributions,J. Math. Pures Appl. 79 (2000), 901–917.
Zygmund, A.,Trigonometric Series I & II, Cambridge Univ. Press, Cambridge, 1959.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Toland, J.F. Continuity and differentiability of Nemytskii operators on the Hardy space\(\mathcal{H}^{1,1} (T^1 )\) . Ark. Mat. 39, 383–394 (2001). https://doi.org/10.1007/BF02384563
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02384563