Research Article

### Interpolation between weighted Lorentz spaces with respect to a vector measure

Year 2019, Volume 48, Issue 6, 1590 - 1600, 08.12.2019

### 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.

### References

• [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.
Year 2019, Volume 48, Issue 6, 1590 - 1600, 08.12.2019

### References

• [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.

### Details

Primary Language English Mathematics Mathematics Maryam MOHSENİPOUR This is me Hakim Sabzevari University 0000-0002-1890-0280 Iran Ghadir SADEGHİ This is me (Primary Author) Hakim Sabzevari University 0000-0002-3815-3681 Iran December 8, 2019 Year 2019, Volume 48, Issue 6