Iterated grand and small Lebesgue spaces

• Autores: Giuseppina Anatriello
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 65, Fasc. 2, 2014, págs. 273-284
• Idioma: inglés
• DOI: 10.1007/s13348-013-0096-1
• Texto completo no disponible (Saber más ...)
• Resumen

  • The norm of the grand Lebesgue spaces is defined through the supremum of Lebesgue norms, balanced by an infinitesimal factor. In this paper we consider the spaces defined by a norm with an analogous expression, where Lebesgue norms are replaced by grand Lebesgue norms. Without the use of interpolation theory, we prove an iteration-type theorem, and we establish that the new norm is again equivalent to the norm of grand Lebesgue spaces. We prove that the expression involved satisfy the axioms of Banach Function Spaces, and we find explicit values of the constants of the equivalence. Analogous results are proved for small Lebesgue spaces.

• Referencias bibliográficas
  • Bennett, C., Sharpley, R.: Interpolations of Operators. Academic Press, San Diego (1988)
  • Capone, C., Fiorenza, A.: On small Lebesgue spaces. J. Funct. Spaces Appl. 3(1), 73–89 (2005)
  • Capone, C., Formica, M. R., Giova, R.: Grand Lebesgue spaces with respect to measurable functions. Nonlinear Anal T.M.A. 85, 125–131 (2013)
  • Carozza, M., Sbordone, C.: The distance to in some function spaces and applications. Differ. Integral Equ. 10(4), 599–607 (1997)
  • Cobos, F., Kühn, T.: Extrapolation results of Lions-Peetre type. Calc. Var. doi:10.1007/s00526-013-0602-z (2013)
  • Di Fratta, G., Fiorenza, A.: A direct approach to the duality of grand and small Lebesgue spaces. Nonlinear Anal. Theory Methods Appl. 79(7),...
  • Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51(2), 131–148 (2000)
  • Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem for weighted grand Lebesgue spaces. Studia Math. 188(2), 123–133 (2008)
  • Fiorenza, A., Karadzhov, G.E.: Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwen. 23(4), 657–681 (2004)
  • Fiorenza, A., Krbec, M., Schmeisser, H.J.: An improvement of dimension-free Sobolev imbeddings in r.i. spaces (2012)
  • Fiorenza, A., Mercaldo, A., Rakotoson, J.M.: Regularity and comparison results in grand Sobolev spaces for parabolic equations with measure...
  • Fiorenza, A., Mercaldo, A., Rakotoson, J.M.: Regularity and uniqueness results in grand Sabolev spaces for parabolic equations with measure...
  • Fiorenza, A., Rakotoson, J.M.: Petits espaces de Lebesgue et quelques applications. C.R. Acad. Sci. Paris, Ser. I 334, 23–26 (2002)
  • Fiorenza, A., Rakotoson, J.M.: New properties of small Lebesgue spaces and their applications. Math. Ann. 326, 543–561 (2003)
  • Greco, L.: A remark on the equality det Df Det Df. Differ. Integral Equ. 6(5), 1089–1100 (1993)
  • Iwaniec, T., Koskela, P., Onninen, J.: Mappings of finite distortion: monotonicity and continuity. Invent. Math. 144, 507–531 (2001)
  • Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal. 119, 129–143 (1992)
  • Kokilashvili, V., Meskhi, A.: Trace inequalities for integral operators with fractional order in grand Lebesgue spaces. Studia Math. 210,...
  • Opic, B.: Continuous and compact embeddings of Bessel-potential-type spaces. Oper. Theory Adv. Appl. 219, 157–196 (2012)
  • Rafeiro, H.: A note on the boundedness of operators in Grand Grand Morrey Spaces, arXiv:1109.2550