SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2019 / Volume: 48 Issue: 6

Interpolation between weighted Lorentz spaces with respect to a vector measure

Maryam Mohsenipour(Hakim Sabzevari University)0000-0002-1890-0280
Ghadir Sadeghi(Hakim Sabzevari University)0000-0002-3815-3681
Pages
—
Publication Date
2019-12-08
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund spaces.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties.
📊
Key Findings & Data:We describe the interpolation with a parameter function of these spaces.
💡
Conclusion & Impact:As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund spaces.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Lorentz spaces · interpolation · vector measures

Abstract

In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties. We describe the interpolation with a parameter function of these spaces. As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund spaces.

🔖

Cite This Article

Mohsenipour, M., Sadeghi, G. (2019). Interpolation between weighted Lorentz spaces with respect to a vector measure. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 6. https://doi.org/10.15672/hujms.652608

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (28)

  1. R.G. Bartel, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7, 289-305, 1955.
  2. C. Bennett and R. Sharply, Interpolation of Operators, Pure Appl. Math. 129, 469 pages, Academic Press, 1988.
  3. J. Bergh, A generalization of Steffensen’s inequality, J. Math. Anal. Appl. 41, 187- 191, 1973.
  4. J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer-Verlag Berlin Heidelberg, 1976.
  5. M.J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112, 480-494, 1993.
  6. G.P. Curbera, Operators into L1 of a vector measure and applications to Banach lattices, Math. Ann. 293, 317-330, 1992.
  7. G.P. Curbera, When L1 of a vector measure is an AL-spaces, Pacific. J. Math. 162, 287-303, 1994.
  8. G.P. Curbera, Banach space properties of L1 of a vector measure, Proc. Amer. Math. Soc. 123, 3797-3806, 1995.
  9. G.P. Curbera and W.J. Ricker, Vector measures, integration and application, in: Positivity, 127-160, Birkhäuser Basel, 2007.
  10. J. Diestel and J.J.Jr. Uhl, Vector Measures, Math. Surveys Monogr. 15, 1977.
  11. R. del Campo, A. Fernandez and F. Mayoral, A note on real interpolation of Lp–spaces of vector measures on δ–rings, J. Math. Anal. Appl. 405, 518-529, 2013.
  12. R. del Campo, A. Fernandez, A. Manzano, F. Mayoral and F. Naranjo, Complex interpolation of Orlicz spaces with respect to a vector measure, Math. Nachr. 287, 23-31, 2014.
  13. D.E. Edmunds, P. Gurka and B. Opic, Sharpness of embeddings in logarithmic Bessel- Potential spaces, Proc. Roy. Soc. Edinburgh Sect. A. 126A, 995-1009, 1996.
  14. A. Fernandez, F. Mayoral and F. Naranjo, Real interpolation method on spaces of scalar integrable functions with respect to vector measures, J. Math. Anal. Appl. 376, 203-211, 2011.
  15. A. Fernandez, F. Mayoral, F. Naranjo, C. Sáez and E.A. Sánchez-Pérez, Spaces of p-integrable functions with respect to a vector measure, Positivity 10, 1-16, 2006.
  16. A. Fernandez, F. Mayoral, F. Naranjo and E.A. Sánchez-Pérez, Complex interpolation of spaces of integrable functions with respect to a vector measure, Collect. Math. 61, 241-252, 2010.
  17. A. Fernandez and F. Naranjo, Rybakov’s theorem for vector measures in Fréchet spaces, Indag. Math. (N.S.) 8 (1), 33-42, 1997.
  18. I. Kluvanek and G. Knowles, Vector Measures and Control Systems, Note Mat. 58, 1975.
  19. D.R. Lewis, Integration with respect to vector measures, Pacific. J. Math. 33, 157-165, 1970.
  20. D.R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16, 583-599, 1973.
  21. W.A.J. Luxemburg, Banach function spaces, Ph.D. Thesis, Delft Institute of Technology. Assen, Netherlands, 1955.
  22. L. Maligranda and L.E. Persson, Real interpolation between weighted $L^p$ and Lorentz spaces, Bull. Polish Acad. Sci. Math. 35, 765-778, 1987.
  23. C. Merucci, Applications of interpolation with a function parameter to Lorentz Soblev and Besov spaces, in: Interpolation Spaces and Allied Topics in Analysis, Lecture Notes in Math. 1070, 183-201, Springer, Berlin, Heidelberg, 1984.
  24. S. Okada, The dual space of $L^1(\mu)$ for a vector measure $\mu$, J. Math. Anal. Appl. 177, 583-599, 1993.
  25. L.E. Persson, Interpolation with a parameter function, Math. Scand. 59, 199-222, 1986.
  26. E.A. Sánches Pérez, Compactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces, Illinois J. Math. 45, 907-923, 2001.
  27. G.F. Stefansson, $L^1$ of a vector measure $\mu$, Le Matematiche. 48, 219-234, 1993.
  28. G.P. Curbera, Operators into L1 of a vector measure and applications to Banach lattices, Math. Ann. 293, 317-330, 1992.