Non-absolutely convergent integrals and singular integrals

• Autores: Petr Honzík, Jan Malý
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 65, Fasc. 3, 2014, págs. 367-377
• Idioma: inglés
• DOI: 10.1007/s13348-013-0103-6
• Texto completo no disponible (Saber más ...)
• Resumen

  • We define the packing integral (a kind of non-absolutely convergent integral) with respect to distributions of arbitrary order. Then we show that singular integrals can be interpreted as packing integrals with respect to generating distributions. This allows us to consider singular integrals beyond L^1.

• Referencias bibliográficas
  • Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, vol. 314 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles...
  • Bendová, H., Malý, J.: An elementary way to introduce a Perron-like integral. Ann. Acad. Sci. Fenn. Math. 36(1), 153–164 (2011)
  • Bongiorno, B.: The Henstock-Kurzweil integral. In: Handbook of Measure Theory, vol. I, II, pp. 587–615. North-Holland, Amsterdam (2002)
  • Calderon, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)
  • Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)
  • Denjoy, A.: Une extension de l’intégrale de M. Lebesgue. C. R. Acad. Sci. Paris Sér. I Math. 154, 859–862 (1912)
  • Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)
  • Henstock, R.: Definitions of Riemann type of the variational integrals. Proc. Lond. Math. Soc. 11(3), 402–418 (1961)
  • Kuncová, K., Malý, J.: Non-absolutely convergent integrals in metric spaces. J. Math. Anal. Appl. 401(2), 578–600 (2013)
  • Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7(82), 418–449 (1957)
  • Malý, J.: Nonabsolutely convergent integrals with respect to distributions. Ann. Mat. Pura Appl. (4) (2013). doi:10.1007/s10231-013-0338-6
  • Perron, O.: Über den Integralbegriff. Sitzungsber. Heidelberg Akad. Wiss. A16, 1–16 (1914)
  • Pfeffer, W.F.: Derivation and Integration. Cambridge Tracts in Mathematics 140. Cambridge University Press, Cambridge (2001)
  • Saks, S.: Theory of the integral. Second revised edition. English translation by L.C. Young. With two additional notes by Stefan Banach. Dover...
  • Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University...